Sadly, there is no secret to it... practice for a very long time with a metronome doing 1234 1324 134 whatever, speeding up as you manage quarter notes comfortably at each speed on the met... and make sure you stay alternate picking when moving between strings. Practice playing 3, 4 and 6 notes a beat... It'll take you a while to get up to speed, just make sure you don't go faster until your accurate at the speed you're doing... muscle memory.

Tremolo picking is just going as fast as possible, but only on one note really and shifting around - it doesn't matter how many times you play a note - think 'Miserlou'.

Gilbert does string skip arpeggios, and you can sweep them as well but it takes practice to lift the pick out of the way of the strings you're not playing - if you're going quick you can mute the others and you'll hear a slight click but that's it. To be honest, I'd worry about one thing at a time. Maybe set aside half an hour a day (or more if you can) to just technique - do alt picking one day and sweeping the next and so on. They all take alot of practice to get right at speed, and there is no point in rushing it if you wanrt it to be clean.... I wouldn't worry about sweeps with string skips until you're comfortable sweeping 6 string arpeggios of various types...

Tremolo picking is like the thing you practice before alternate picking. You should be able to alt pick as fast as you can tremolo pick. Trem is only your picking hand, alt requires co-ordination (obviously!)...

Good luck man.... it is a pain in the ass, but once you're there, you only have to keep it up... there are lessons on this site, and pretty much everywhere else to help you along...