Leaded rosin-core solder flows nicely, but you should be fine with the solder you have. Don't you have a hobby store of some kind nearby? Those generally have good 30-AWG wire. If not, there's always eBay and DealsExtreme and friends. For example, this (you'll get 50 feet, not 5 spools, but it's still a decent price, and way more than enough wire for fixing a few calculators).
I can't really tell from your pictures but it looks like the solder joints are quite rough and dull rather than smooth and shiny. As you say you're having problems getting the solder to stick my best guess is you're not heating the pad/wire enough, but spending too long trying to get the solder to stick and overheating the solder.

As the metal surfaces heat up they oxidise, which prevents the solder from sticking. The solder contains flux to remove the oxide layer, so you need to keep feeding fresh solder to your soldering iron to maintain a good supply of flux to keep the surfaces clean and oxide-free.

You need to make sure both metal surfaces (pad and wire) are clean and hot before applying the solder - don't hold the wire to the pad and try to "paint" molten solder onto it, as this will form (at best) a brittle/"dry" joint.

Make sure that the wire is held in position against the pad - use your helping hand or tape the wire to a nearby surface to ensure it's not going to move away when you apply or remove the soldering iron. Tin the soldering iron bit (apply a small amount of solder then wipe off the excess with a damp - not wet - sponge, so you have a nice shiny bit) then hold this against the pad and wire for a moment to heat them up. With the soldering iron in position apply the solder to the joint then remove the solder and iron in one movement, then allow the joint to cool - it should be smooth and shiny. Clean and re-tin your soldering iron frequently to maintain a shiny tip.

One thing to watch out for is not to overheat the pads on the PCB - if you hold the soldering iron against them for too long they can become unstuck and lift away from the board. Lead-free solder has a higher melting point than lead solder, which means that irons have to be hotter to compensate - this results in faster oxidation and makes it easier to damage parts through overheating.

This may well be repeating what you're already doing, but without seeing how you're currently working it's difficult to suggest anything - the above works for me, at least!
My brother and I just got a new soldering iron, and I was elated to discover it functioned vastly better than my old one, even to the point of making me wonder how I ever got on at all before! My old one was 25 watts and the new one 15 watts, but for all that, mine took forever to heat a joint if it did at all, and the new, about a second. Mine also felt much hotter when I put my hand above it, like a stove burner, but the new emits very little heat that way. Cool beans.
this is a nice project you've got Smile
may i ask where you got that soldering iron??

off: that's a cool quote in your sig. KermM, do you mind if i use it on other forums aswell?
Legodude, no problem. Caleb, glad to hear you got a better soldering iron, and I hope it helps you complete your projects faster and better. Does that mean you'll be picking up this project once more, since you posted about the soldering iron in this topic?
Oh yes, this project isn't dead, only waiting for some smaller wire. Not easy to obtain, by any means! Now, it doesn't mean the calc will work; something might have gone wrong, like I might've burned the surrounding area up. But I shall try to complete the soldering.

Legodude: Which iron: The old was from Wal-Mart, and the new from RadioShack, the $20 "Pro-Line" 15 watt model.
hmm, sucks. i live in the netherlands, we don't have radioshack Sad i'll need to search for it on the internet.
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