When you are dealing with plants which are going to end up in containers anyway you absolutely could opt to seed them into those containers indoors and then later move the containers outside once you've hardened off the plants. They might turn out just fine that way and I'd absolutely encourage you to try that - seeing what works is part of the fun of growing things.
Often folks - and I've done this myself - initially put seeds into trays or peat pots or some sort of biodegradable container and put them under a grow light or just a fluorescent bulb, typically inside some sort of covered tray to help keep the environment moist and a bit warmer. I've put trays atop my fridge before to do this. This isn't always possible to do with the container that they'll eventually end up in. Additionally, when you plant them in a relatively shallow tray, it is easier to keep the seedlings in a moist planting medium. Often in a larger container, much of the water works its way to the bottom. Plus, the mass of potting soil doesn't readily heat up as well as the more shallow tray. Seeds need moisture and warmth and then later some sunlight (or a reasonable facsimile - e.g., fluorescent lighting).
The key with transplanting seedlings is to make sure you've got a well-established root system, have your new planting hole ready to go and then get that seedling moved w/out having all the planting medium falling off those roots. If you bring enough of that planting medium with you when you put it in the new container the plant won't really "know" that it's been transplanted.
Also, when transplanting and I find the seedlings are root-bound, I'll take a blade and lightly cut a bit on the root system (just slash a bit on it along the bottom) to help encourage better root growth.
Hardening off, though, is a key step in successfully moving a plant that has been inside to its new outdoor environment. That just involves putting it outside progressively more over maybe a week to ten days. Seedlings inside haven't been dealing with lots of sunlight (presumably) or the wind or the cooler night temperatures or the less-frequent watering that tends to happen when they go outside.
When I've transplanted seedlings outside I'll start off by isolating them from the wind but give them sunlight for a few hours and then bring them back in. Each day I'll increase how long they spend outside and cut back a little bit on the watering. I don't do any fertilizing or anything like that during this period. I watch the weather and bring them in if it is going to to be severe.
Well, I've wrote a book. Hope that's helpful.