Timely topic. Let me say in summary: it is more about personal gratification and happiness than it is cost savings...
I have done this off and on through the last 30 years. I found that the various systems to start in a peat medium work pretty well. Here is just one example of many:
https://parkseed.com/park-bio-dome-seed-starter/c/bio-dome/
However, it really depends on the species. For about 15 years, here in the South, I was EASILY rocking Impatiens using this system. It was very cost effective.
And then Armageddon hit.
Downy Mildew arrived in the early 2010s.
I'm not joking, it is Armageddon. If this has not hit you up north, then go for it. But here, impatiens are a lost cause. I used to have mounds that would grow to 3 ft high and 3 ft wide from just a few plants. No more.
I switched to wax begonias. They are also fairly easy to start and I recommend them. Not as easy as impatiens, but they work OK. But you need a really early start.
Geraniums are a no-go here, so I have no experience. Can't comment.
I CAN comment on New Guinea Impatiens. They are
super tricky. I found them almost impossible to start from seed. Maybe 1 in 10 worked for me? This is versus 100% for impatiens and begonias.
I also find coleus to be very, very easy to start by seed. Nice for shade.
I also have gone with a few species you can just scatter seed with. These are the sun loving marigolds and celosia.
Now, all that said... This year we were out of town in late winter, so I skipped it. I planted store bought begonias. Guess what? My compost was not hot enough over winter and it carried last years coleus and celosia with it. I had a nice raft of "volunteers" which I intentionally let grow and crowd out the begonias. I also picked off a few volunteer tomatoes and grew them for a "what the heck" factor. Turns out we got 100s of cherry tomatoes from that free crop.
Basically, don't forget your volunteers from last year's crop!
Here's a picture of my volunteers (celosia and coleus, crowding out my begonias). Just incredible. And free.