When it comes to removing solder, you must get all of the solder melted first. Make sure you use solder flux on both sides of the board...it will ensure that the solder gets melted and flowing. If you missed getting some of it out, you can try reflowing some fresh solder onto it and then try sucking it out again. Most of the better shops will have a desoldering station which has a gun with a hollow tip and a vacuum pump...once you get the solder melted, hit the foot switch and poof, the solder gets sucked out. If the 25 watt gun is not melting all of the solder on both sides of the board, then try going with the 35. A lot of guns have the capability of switching to higher wattages.
Concerning the solder wick (stuff that absorbs solder,) it can cause more problems than it's worth. There are a few of the old die hards that still continue to use it but I strongly recommend staying with a good solder sucker such as your spring loaded model or using the one with a vacuum pump attached.
When you go to install the new component, again make sure you use that solder flux. Once you're finished, clean all of the flux off with isopropyl alcohol and an acid brush. The flux is highly corrosive if left on.
Just a few pointers...Good Luck!