Why does my engine die after stalling? Manual transmission?

So I recently purchased a 93 Honda Del Sol, with 125k miles, and I bought it with a manual transmission because I wanted to learn it in a bucket so I could then transfer over to a newer model also stick. So as I'm learning I tend to stall the engine on that first gear, and half the time when I try to start it back up the engine won't turn on, or it ll turn on and I put in gear and the engine just dies out. But the battery is fine, electronics work, the car does crank up but won't turn on. But I leave it off for about ten seconds and then the car starts fine, and I'm able to "drive off", cause I honestly accelerate too much and let go of clutch to get me rolling. But yeah what could that problem be? Could my starter be messed up? Or is it the engine? Could I possible just need a tune up?

I'm not going to jump to conclusions on this one, but a common problem with Hondas of this vintage is bad fuel system relays. Get your car diagnosed.