I had a nightmare late last year, and got buckets of advice from people on this site, so it's only fair that I lob some back in.
1- Schwalbe Marathon+ seem to work pretty well. They're not slicks so aren't as quick as some other tyres, but they have grip so are better in the wet. And they're hard to puncture.
2- When you put your tyre on, line a bit of the logo up with the valve. It helps you to identify if there's something lodged in the tyre. I had a bit of glass stuck in the M+'s that was puncturing the tube every 8 or 10 miles.
3- Stay further out from the gutter at the edge of the road.
4- Inflate your tyres properly
5- When you've had a few holes put in an inner tube, it's probably worth replacing it.
6- Carry spare tubes. If you patch by the road side and re-inflate, you sometimes find the patch doesn't stick properly.
7- Stick at it. It's worth it.
