Heres some tips from me:
Relax, keep everything flowing. Dont rush up to things when you first start out. Ride along relaxed taking everything in.
Dont just leg brake at the start. Slowly start leg braking more as you go along. You might find if you just try leg braking to slow you down alot your knees might ache abit. Slowly bring it in.
When its sunny and your bored, go have some fun on your fixed, cruise around on it, go to an empty car park and have some fun. Have a go at trackstanding (very usful skill and look pretty cool), backwards pedling and riding around slowly. Do some skids aswell if you dont mind, that will get you used to how it feels when you stop peddling and should make you less scared of it. Also pretty cool i think

Do some riding around slowly and relax your legs so you can feel the pedals pushing your legs around. This will help with going downhill.
Go down some hills and get used to pedaling fast, dont go too steep too soon. Find a nice hill, and when you get used to spinning down it try and spin faster down it
This should help you abit and get you used to fixed.
Oh, and when you start off, just push down, get onto saddle and relax that leg so it moves around. You shouldnt be going too fast and you should be able to clip in with the other foot when it comes up, if not then keep relaxed and try again.