Smileyman hit the likely culprits. Definitely check the connections at the clutch switch. Unplug it and plug it back in to see if that helps. I'm thinking a bad sidestand switch would probably prevent the push start, too, but it can't hurt to check that connection.
When you get the meter, find the starter relay and see if you get power to it when you press the start button. If not, it could be a bad start switch. If you get power to the relay, but no power past it, it's a bad relay. If the power's getting to the starter, but nothing happens, it's a bad starter. If you're lucky, the solenoid is replaceable separately. If you're unlucky (like my dad), you gotta replace the starter.
If you don't want to bother with the testing, just throw a starter relay in there and see if it helps. They're pretty cheap.
Edit: Never mind the switches then...
If it's not the battery, it's still worth checking the relay. They can do some freaky stuff when they're going bad.