The first things you should do are these:
- For long-lived and/or expensive perennials, in cold areas, I personally recommend looking for plants that can survive hardiness zones that match the record low temperatures for your area. Hardiness zones are based off of average coldest temperatures in an area, however, which explains why they might call an area zone 7a that can get down to -30° F. once every 15 years. I don't recommend planting such as fruit trees for hardiness zones based on average temperatures, however (but I'm sure people do it, and some of them probably have great success); however, they probably won't last more than 3 to 5 years, in some areas (depending on the area and the plant). Use this map to find the temperature ranges for each zone. However, I recommend finding out how cold your area is yourself, via searching for weather data, asking locals, etc. (not by looking at the map); my area is charted for zone 6b or 7a at that link, but we plant fruit trees for zone 4, because it's not uncommon for it to get significantly colder than -5° F., and on uncommon years it gets -20 or lower. When you buy perennials, they often say a hardiness zone range for them (that is useful information). We've lost very few trees to the cold (if any at all) since we started planting for zone 4. And, just because they'll survive in your zone doesn't mean they'll fruit in it (another reason to buy down a zone or three sometimes). Frost in my area can kill at least some kinds of apricot blossoms pretty easily, but the trees survive the winters well; they just bloom early for the area is all.
- For short-lived perennials (or ones that are easy or inexpensive to replant), it's probably not going to be a big issue if you take more risks, if you want something that needs somewhat warmer temperatures.
- For all crops (especially annuals), find out how long your frost-free growing season is. I know you mentioned when it started to get warm, but I'm not sure if you factored in the frost.
- For indoors, if you're not growing under lights, you're going to have to do considerable research and experimentation (beyond any tips we have to offer). There's not a lot of data on which fruiting plants do the best in those conditions (although some plants definitely do better).
- Find out how cold-hardy your annuals are (peas are more cold-hardy than peppers and melons, for instance). So, you can plant peas a lot earlier in the spring, but a lot of crops can't be transplanted until after the last frost (unless they're protected with cloches, row covers, or something).
- Look at the days to maturity (dtm; days) for the seeds/plants you buy (for your outdoor garden). You'll probably want early varieties, for the most part (but if you don't mind a smaller and/or much later harvest, you can still plant late tomatoes and stuff like that).
- Realize that some plants have fruits that don't taste as good when it's cold, and some taste awesome then (i.e. these Tatura tomatoes tasted great at the end of the season for me, in 2017; some tomato varieties have been another story). It will probably take some research and experimentation on your part to determine which ones taste the best in your area.
- When you start your seeds, as already noted in another answer, you'll want to start some things early. I like to start mine in an unheated Strong Camel 6'x5'x3' greenhouse (outside), up to a couple months before the last frost (our last frost is usually about May 10th). Yes, that does work, even in colder zones, even though it's unheated, but there are some tricks to know (like your plants probably won't sprout very fast, if at all, if you don't open any vents). I prefer the greenhouse to starting indoors because the plants get a lot more light, and the light is free (plus it only takes up outdoor space—not my bedroom); the plants are stronger and more used to outdoor conditions, too. But, starting indoors can work, too, of course (and you can even have success without grow lights, especially if you have a good south window—but more light is ideal, IMO).
- Realize that there's more you'll need to know than what plants can grow in areas that get as cool as yours. You may also want to make sure you plant plants that are well-suited to your soil type. For instance, if you have clay soil, you'll have a harder time finding the right plants for your garden, perhaps, since there's a lot to know (but it's not a lost cause): For clay, you'll probably want to plant wider, shorter carrots (as opposed to long ones). You'll want acidic soil for potatoes and blueberries. Anyway, those are just some examples of some things you'll probably want to look into. Soil pH and soil type are important factors.
- I would recommend taking a trip to your local seed stores and see what they sell. If they sell it, there's a good chance you can grow it. Stores that aren't locally owned might occasionally carry seeds for things that don't grow in your zone (but most of the stuff should work well).
You might try plum and pear trees, and apples. Again, check the zones.
Perennial non-tree plants:
- garlic chives
- bunching onions (I like Crimson Forest, so far; they multiply every year; they grow tall greens, they taste great, and they're easy to grow from seed)
- potato onions
- sorrel (this is like perennial spinach, but a lot more sour than spinach)
- strawberries (alpine strawberries can even have fruit long after the frost, apparently; I saw some out there recently; it's almost December! Alexandria is a good variety to try.)
- black raspberries
- garden sage
- service berries
- bilberries (give them acidic soil)
- blueberries (give them acidic soil)
- potatoes (they're perennial if you don't dig them up)
- sunroots (AKA sunchokes / Jerusalem artichokes)
- Sempervivum tectorum (it's edible, but most people grow it as an ornamental)
- hard neck garlic
- dandelions (yes, some people eat them; they're also a herb)
Annuals biennials and such:
If you get the right varieties, you should be able to grow most common annual vegetables and herbs. Here are a list of things (not comprehensive) that could probably work (although there are ins and outs to many of them); although you probably can grow all of these, you might need specific breeds or special attention in some cases for it to work out very well (I'll mark things with an asterisk that I think should be easy for beginners):
- If you grow regular onions, get long day or intermediate onions. Short day onions are for the south (and may not bulb in the north).
- Zucchini (you'll need to make sure they get enough sun, and hopefully enough heat in the summer)
- Tomatoes (e.g. Sub Arctic Plenty, Early Girl F1, Sasha's Altai, Sweet Orange Cherry, SunGold F1, etc.)
- Peppers (you'll really want to find ones that can handle the cold; King of the North is supposed to do well—although it was late in my area, this year; I've heard that Ace, New Ace, Carmen, and Hungarian Wax worked well for a New Hampshire gardener, who had tried growing different pepper varieties for over thirty years)
- Tomatillos (I think in colder areas, the plants do better if you let them reseed)
- Ground cherries (I think in colder areas, the plants do better if you let them reseed.)
- Melons (You might like Minnesota Midget :); I don't know if your area is hot enough in the summer to be a good fit for most melons, but I know at least some northern areas, like mine, have hot summers)
- Watermelons (You might look for early Russian varieties and stuff like that, since they should be used to colder-than-average areas. Blacktail Mountain is one of the earliest watermelons; I think I got my Moon and Stars watermelon seeds from someone who grew it in an area not far from yours, but that is a rather late variety compared to my earliest ones. Verona is earlier than most, too, but later than Blacktail Mountain.)
- Basil (Spicy Globe Bush did very well for me, this year; it's very prolific all season and tastes great on pizza)
- If your summers get hot, I recommend Armenian Cucumbers
- Okra (this might sound like the last thing you'd want to plant in Minnesota, but if you start it early and transplant it, it should be able to do better than most peppers will—at least, they did well for me that way; they didn't do well at all direct-seeded; okra seems to be the first thing to sprout in my unheated greenhouse, too; it takes a while to mature to a size fit for transplanting, though; I recommend Edna Slaton's Candelabra; it's early and the pods are still tender when they're very long; I think I like Bowling Red best for taste, though, so far)
- Wonderberries* (I like these a lot; very easy to grow; they're early and produce all season, too; they have small fruit, however. You might still want to start them early, though.)
- Etc. (I may add to, amend, and/or clarify this list over time.)
Plants that can do better than average (and fruit) indoors, in a south windowsill, without growlights (I'm not going to get into growing under lights much—but gynoecious, parthenocarpic greenhouse cucumbers might be great for that):
- Candlelight peppers
- Ground cherries
- Grandpa's Home peppers
I think Candlelight peppers have been the most fruitful plants I've seen in a windowsill, so far. Your soil and container size is going to be important for plants that can actually handle low light. Peppers can require a lot of care, though; you might have fewer nutrient and disease issues with ground cherries (but I could be wrong).
I've grown Grandpa's Home, but not indoors; it's supposed to be shade-tolerant enough to make a house plant (so, that's why I included it).
Ground cherries seem to fruit indoors as a matter of course. I only had them in cups at the time I got fruit on them indoors, but if you had them in a decent container, they might give you a lot more fruit (they fruit a lot more indoors than the tomatillos and tomatoes I've tried do). Pollination doesn't seem to be a big issue for ground cherries. Aunt Molly's was the kind I tried, but I wouldn't be surprised if other varieties were similar indoors.
Some tomatoes will fruit better indoors than others, but I haven't found any that fruit a lot indoors. Maybe Red Robin is a good one for that, but I don't know.
You might try other lesser-known Solanaceae berries indoors to see what happens. Ideal plant varieties and species probably aren't going to get very long or tall, since those will probably require much larger pots, and more window space.
Anyway, my area probably isn't quite as cold as yours, especially during the summer (it can be really hot and desert-like here in the summer, which is great for muskmelons). Our growing season is usually about five months; from the sound of it, yours is similar (maybe somewhat less). I'm in Idaho.
Joseph Lofthouse is a landrace gardener who gardens/farms in a very short season. He sells a lot of innovative, adapted plant breeds for various species that can do well with a shorter season. (Both Joseph Lofthouse's area and my area are dry, probably with higher elevation than yours; his area is cooler than mine, with a shorter season; elevation is supposed to matter for some plants, but I'm not sure what to say about it). Anyway, he grows all kinds of landraces.