Good advice above.

Will definitely second the patch kit. Yes, tubes are 'cheap', but every once in awhile I have similar episodes of repeat flats right after mounting, and bam- there's $14 gone down the drain without a patch kit.

I wait until I've got 5-6 tubes ready to restore, and do 'em all at once. It's pretty fun, actually, aside from the real money savings.

There's a counterintuitive but helpful tip on the patch kit - have a small knife or screwdriver on hand to make the puncture hole BIGGER so that you can easily see it. Sounds weird, but your biggest problem with the patch will be putting the patch dead center on the puncture. The holes are usually small enough that uninflated, it's barely visible. Widen it to a good 1-2 mm, and you'll drop the patch dead center every time, and the fix lasts forever.