That's hardly an upgrade... If you spent less than $10, you have a pretty minimal soldering iron. They're OK for quick one-off jobs and infrequent usage, but if you actually plan on using the thing frequently, get a decent one. I'm a fan of Weller simply because the tips are readily available and work well, but you can get full digital temp controlled setups from no-name brands online for like $50-75.
As for tinning the tip of cheap irons, there's a fair bit of patience and luck involved. Let it get good and toasty, at least 325 celcius (which you can't measure easily, but oh well), then flow some solder onto it. Most won't stick. Wipe it off on a damp (not wet) sponge, let it heat up a little again, then repeat, working your way around the tip using the edge of what "took" the last time. With luck, you'll get most of the tip tinned. If you're unlucky, the flux will eat away the tip too badly before it actually takes and you'll have to start over with a new tip. The Radio Shack tips tend to at least work reasonably if you're patient, while the really cheap ones never seem to take.
Radio Shack does have a tip cleaner/tinner product that I'm told works well. Only use this stuff on the cheap steel tips, though. Higher end tips like those used by Weller and Metcal are coated in a separate metal layer that will immediately take solder well, and tip cleaner (chemical or mechanical) can destroy that.