It is my first time planting anything from seeds, and certainly my first time with cherry tomatoes. I've watched many tutorials and read many articles, so I have a fair knowledge about the subject. What I'm not sure of is whether or not the sunlight from my window will be strong enough for the seedlings to grow. Here's a picture of the window where the plants will be.
Seeds don't need light to germinate, they just need moisture and heat. But they need light to grow. Fruiting plants need the most light of all edible plants as the energy from sunlight is needed to create the carbohydrates etc that form the bulk of the fruit. Vegetables that are only leaves such as lettuce, and root vegetables, need less light.
So, unless you get a minimum of 6 hours of direct sunlight through that window shining on the leaves, you'll need to add grow lights.
Unfortunately, that level of light is probably not direct or strong enough to support the seedlings. Even if it was, seedlings reach for the light, so they'll not only be leggy, meaning with tall but skinny stems, but they're liable to actually break as they bend towards the window. I've had it happen, and it's sad to see things germinate, only to watch the new babies break in two and fall over.
It's hard to tell how much light you're getting. So, it's hard to say. Here's my recommendation:
- Get some shade-tolerant tomatoes. There are at least a few out there, but they might not be ideal for your purpose. If they're heat/cold/drought tolerant, and parthenocarpic, that's even better, but I'm not sure if any one tomato meets all of that criteria, yet.
- Give them plenty of soil. I find that outdoor soil encourages fruiting on indoor plants more, but it may have diseases or pests that may stick around in your house and infest your houseplants, too. (So, I don't recommend outdoor soil particularly.)
- Learning about plant regulators (plant hormones) may help. Some are supposed to help prevent blossom drop. (Which tends to be a problem indoors).
Really, though, I'm guessing you could do with some grow lights. Just get some CFLs or something. Make sure some of them are lower color temperatures so as to get enough red light (which is important for flowering). Higher ones (which is what people tend to recommend) help mostly for leaves.
I would recommend trying Grandpa's Home peppers instead of cherry tomatoes, personally. They're supposed to do well on limited light, and are said to be usable as houseplants.