Do you kickstart it while sitting on it?
Don't wanna sound like a wuss here, but when I first got my 72 Trophy I stood next to it to kickstart it. The guy I bought it from showed me that's how he usually started it. You might give that a go until you get the hang of it.
As the others said, make sure to put it in neutral, pull in the clutch, then work the kickstarter a few times to break the clutch plates free inside the gearbox. You'll feel a little resistance at first, but once you hear that nice ratcheting sound you're ready to let out the clutch handle, give it a tickle, turn the key and fire it up.
And watch out for the kickstarter to kick back. It got my ankle once when I wasn't wearing boots.
You may not always need to use the choke. If it's warm out I doubt you'll need it. Some people find the open/closed positions on the choke counterintuitive. Here's the scoop: you don't want to drive around with the choke pushed forward, it should usually be pulled back.
I generally turn my headlight off while starting the bike, just to give that extra bit of juice.
Make sure to turn the fuel petcocks off
every time you park it.
If you didn't get an owner's manual, get one here and check page 6 of the pdf (page 11 of the manual):
http://www.classicbike.biz/Triumph/OwnersManuals/OwnersManuals.htm
And I usually kickstart it while sitting on it (now). It usually goes first time, unless someone's watching.
hope that helps