ummm... i don't think there's afterburner in interceptor 
There seems to be in the demo. TAB key. Perhaps it was removed in the final game?
anyways, the main weapon against ufos is the missiles/torpedos. just keep the ufo on target until the thing locks-on and fire away, and you'll see the missile go straight after the ufo. depending on the difficulty and ufo size, the ufo will avoid it or not. probably not, if it's the start of the game.
I haven't used missiles much. Didn't realize they're the weapon of choice. Makes sense; you're not going to have more warping in, right? Only so many to deal with. I'll have to break this X-Wing/TIE Fighter mentality.

anyways, if you're going to hit them with beam weapons (specially if you've used up your missiles), it is hard, but not impossible. first, the trick is to not try to go as fast as them or always on their tail, since they easily change directions and you'll hardly have them on radar. either increase shield / weapon recharging (iirc, somewhere between F9 and F12, by default) to use more power and decrease your speed, or try to keep a distance from the ufo. if you get too close just reverse (default: Z key) and then go back to normal (same key). as long as you always have the ufo on screen, you should be able to hit it without too much fuss. beam weapons can hit up to a distance of 1km, i think.
I normally have both weapons and shields at high recharge. I guess I'll try raising them higher.
The Z key I haven't tried yet, I'll look at that too.
btw, what difficulty you're playing in? i'm guessing something high since your squadmates keep going down. my squadmates are always much better than me and nail most of the ufos, which is quite useful in high difficulties. and you can always boss'em around with the proper controls 
Beginner. But again, it's the demo, so things may not be quite the same as the full game if they did it before the full game was released.
btw, i play in Genius since Beginner is insanely easy and Superhuman is insanely hard. in high difficulties ufos have armour and are real tough to hit, while they hit you quite easily, which is BAD 
I get hit often enough as it is.

Probably about a 1/3rd of the demo missions I've done so far ended with my death.
if you are really having trouble, then keep in reverse with a good engine, then you will exploit a bug in the system.
You mean with engines at a high blast, to keep the UFO trailing you and in your sights?
A tip for beam weapons is to aim at the cursor in FRONT of the UFO which guarantees you a hit more or less.
Yeah, I know aiming at where it is when I fire won't do much.

My aim does need work though.