How to solder something as tiny as this?
In looking at your picture, I'd say it looks like to me that:


Your wires are too long (too much insulation stripped)
Your wires don't appear to be adequately tinned
You appear to have stripped the wires with a pocket knife or similar (shouldn't impact solderability, but looks bad and may contribute to #1)
It doesn't look like you twisted the wires together before you tinned them. 


In general, I squeeze wire strippers on the insulated wire to cut through the insulation, then twist the cut end as I pull it off to get a tidy twisted end, like this (though that's a lot longer than I would generally ever expose) 

Once that's done, I coat the neat wire end with a copious amount of flux, then get a gob of solder on the tip of the soldering iron, then lay the gob on the wire until the solder is wicked into the wire. No need to rub the iron on the wire, or press, or do anything else - it'll all get soaked up as soon as the wire gets hot enough. 

Once that's done, keep adding solder until right at the point you can barely distinguish the original stranding. At this point, the wire's tinned. If you tin the pads with a **little* solder, then soldering the two parts together should just involve laying the soldering iron on the wire with the wire on the pad. 

You shouldn't ever need to push or rub or poke with the iron. Be sure to clean the tip off with a damp sponge immediately before you do any work.

If you're looking to get a solder joint like the one below, check out this tutorial on tinning wires and this tutorial on soldering to LED strips.