Saving money by growing your own? I don't think that's possible unless
you work for free.
Can you? Should you? Different questions. Purchasing seeds by
mailorder: seeds are cheap, shipping and handling are not.

You certainly can start plants under lights; I'd recommend 2 shop light
fixtures, that gives you roughly 4 square feet of space. Space the
lights about 1 foot apart, on a system that lets you raise/lower the
lights. I raise/lower my seedlings, same principle.

Lots of space for seedlings, but not so much for plants.

Put lights just above your seedlings - flourescent lights won't burn
your plants. You don't REALLY need gro-lights, but a full-spectrum
tube (one per fixture) would be good. I use regular old tubes in mine.

Dixie cups for small stuff - sure, but you'll have to be careful
watering that you don't over-do. I usually start tomatoe plants in a 3
or 4 inch pot, several per pot. When seedlings have two true leaves, I
put each seedling into its own cell in a 6-pack cell (handle with care,
by leaf only). When I see roots coming out of the bottom, I transplant
to a 4 inch pot. When I see roots coming out of the bottom of that, I
TP to a larger pot. Usually about this time the plants are 12-16
inches tall. I usually put them in the ground too early, but at this
point they make the ones for sale look puny. However, all tomatoe
plants can by transplanted to the garden and buried except for the
tips, and they'll root all along the buried stem.

A dozen plants sounds like a lot Unless you really like tomatoes
that is!

The plants will do fine uder lights; you'll want a timer to turn off
the lights 5-6 hours every day. You also need to take care when
setting your plants out - harden them off over a week. Don't take
plants from your lights and put them into full sun, you'll kill them.
They need gradual exposure to sunlight and weather. Easy does it for
the first couple days.

Usually the main reason for growing your own is variety - plants for
sale are usually just a small piece of the plant spectrum you have
access to by starting your own plants. I don't think I save any money
by starting my own, because my seed bill is usually 30-60 dollars and I
nurse those plants along for 2-3 months. Figure water, electric, time
& energy, space in the house.. Expensive plants. But in my opinion..
worth it all

Good luck!