When I first tried indoor seed starting, my biggest mistake was assuming my south-facing windowsill would be enough - it wasn't. The seedlings got leggy and weak because they were stretching toward the light source. If you want decent results, you really do need supplemental lighting. A basic LED grow light or even shop lights with cool fluorescence will make a massive difference. Keep them about 2-3 inches above the seedlings and run them for 14-16 hours daily. Your windowsill can be part of the setup, but it shouldn't be your only light source unless you're in an extremely bright location and can rotate the trays constantly.
On the soil front, the earlier answers nailed it - seed-starting mix is worth the investment. Regular potting soil compacts too easily and holds moisture in ways that invite damping-off disease, which will kill your seedlings fast. The lighter texture also lets roots establish better early on. I'd add that you don't need anything fancy; a basic seed-starting mix from any garden center works fine.
Beyond those two things, stay consistent with watering. Keep the soil moist but not waterlogged - the seedlings hate sitting in standing water. Use a spray bottle or water from below if you can, rather than pouring from above and disturbing the tender plants. Temperature matters too; most vegetable seeds germinate best around 65-75°F. Once they've sprouted, you can drop it a few degrees if needed, which also helps prevent those stretchy, weak seedlings.