Growing your own tomatoplants from seed is easy. However, it may be easy, it isn't always possible. In that case it is perfectly acceptable to buy some seedlings from a store or find your local facebook exchange group and find someone who has too many. Don't worry about it. It is more important to grow delicious tomatoes than to do all the work.
The problem with growing tomatoes from seed is mainly a logistical one. You need the space and time, and the correct timing.
The more seedlings you grow the more difficult it gets, with the difficulty seemingly increasing on an exponential scale. Taking care of 5 seedlings. No problem. 10. Doable. 30, it is starting to be a lot of work. 200 seedlings: better give up your daytime job. So be warned. Don't try to overdo it. Especially if you have difficulties giving away seedlings, or composting them.
Timing is also an factor. You need to have your seedlings at the right stage when you can plant them to your final position. You don't always get this right and it isn't always your own fault. Seasons aren't always the same each year. Sometimes you have a great spring, sometimes it looks more like winter.
Tomato seed is very sturdy. You have 10 year old tomato seed? No problem. It will probably germinate fine. In fact, you might grow some spontaneous tomatoplants on your compost heap if you trash tomatoes there. Germinating isn't a big problem as long as you give your seeds some growing medium and moisture.
Picking your initial growing medium can be difficult. That's because everybody has their own opinion on the matter. But as long as you don't have a professional tomato seedling operation going on that needs optimizing, any standard growing medium will do. I often use just a general garden compost for starting my tomato seedlings, even though you see people recommend you not to do this. Your seedlings will not be long in their first pot. It won't be optimal, but it will be fine. If you do use general garden compost siff out the rough stuff first. If you don't want to take risks use a growing medium made for seedlings.
I would recommend using plastic pots. The ones from organic material seem to be a nice idea but they drie out much faster than plastic pots. And because the original pots are so small this poses a real theat of dehydration for your little seedlings. You will have to water much more frequently. Not recommended. Self-made pots made from newspaper are a bit better than the store-bought ones. But I would go plastic all the way.
You can sow your seeds in individual small pots or large trays. The key to both methods is to label your locations accurately. Labeling small pots is easier if you have many varieties. Trays are efficient if you need lots of seedlings of the same or few varieties. The most common way to label trays is to label rows.
Fill your pots with the growing medium. Add a few seeds per pot. For instance a seed in each corner. Then cover the seeds with about half a centimeter of growing medium, and water them down. This is usually the last time they need watering the seedlings show themselves. Put them in a warm place and seedlings will pop up within a week to two weeks.
If nothing shows up after two weeks, carefully examine one pot. Try to remove some of the medium and see if anything is growing. If there is nothing, redo your seeding. Be aware that overwatering is not good during this early phase of growth. Seed needs moisture to germinate, and that is about it. If you water them too much during the germination process the seed could very well be rotting instead of growing.
Some people like to put a lot of seeds in one pot. The idea is that once they are seedlings showing their first true leaves you separate them all and plant them into their next pot. If you need a lot of seedlings this is definitely a way to go. It saves a lot of space in the initial phase of the operation. If you are a casual grower, don't bother. Just add 1-4 seeds per pot. Remove seedlings that look inferior.
Your tomato seedlings have shown themselves! Pretty soon they will start making their first true leaves. They start respirating now but do not have a large root system. Because your seedlings are in small pots you will need to keep an eye on the soil moisture. Water regularly. Feel the earth with your finger. Don't overdo it.
When should you transplant seedlings to the next pot? You have two options. You can start transplanting your seedlings to individual pots once the first true leaves show, or when space is an issue wait longer. The longer you wait, the more difficult it will be to disentangle the root systems if you have more than one seedling per pot.
After transplanting the seedlings will put more effort into developing their root system and into the stem and leaves. To help develop the root system you can let the pots go pretty dry once in a while. This stimulates the plant to extend its root system. A well fed and watered plant will have a lazy root system. The seedlings are also a bit more sturdy now and you can actually train them to become stronger. Not by singing to them, although it might helpm but by touching them. You can also gently stroke the top of your seedlings with your hands so that they wiggle back and forth. It starts a physiological process that strenghtens them and makes them less leggy. You can also blow gently. You don't need to do this for long periods of time. Just once in while. You are basically simulating a microclimate and the plants will respond to it.