A soldering kit does not have to be very expensive. The tools shown in the above photo are what I could to any soldering job with. You do not need them all and we will discuss the important ones below.
For capacitor removal on motherboards it is important to have a reasonably powerful soldering iron. A 40w is the minimum that you would require. Actually a 60w is preferred and an 80w is what many of the professionals use. It must be a grounded soldering iron.
Soldering Stand with Sponge
Soldering Irons are available in corded models or as Soldering Stations which are Analog or Digital. For desoldering capacitors from motherboards it is not absolutely necessary to get a soldering station and definitely not necessary to get a digital one because you will be desoldering capacitors at the maximum temperature (450oC or more).
It is important to get a stand for safety reasons though and most important to have a wet sponge on the stand so that you can clean the iron of solder periodically in order to do the best soldering job.
Soldering Iron Tips
You might think that a very thin tip is required for removing capacitors from motherboards. This is incorrect. A thin tip will not get hot enough and transfer enough heat to the board. The best tip to get is a chisel tip. Around 2mm is the best size.
Solder is a matter of personal preference. Standard 60/40 solder is fine but preferably around 0.8mm diameter.
Large clippers are no use for cutting the capacitor leads. You need small lead clippers.
A pneumatic solder sucker is a type of desoldering tool. It is useful for removing excess solder or desoldering damaged connectors. It is not recommended to use it to clear the thru-holes in the board because the pressure is excessive and there is a chance that when it recoils it may hit the board and damage a trace. If you insist on using it to clear thru-holes of solder then use it half-cocked.
Antistatic Wrist Strap
It is not necessary to buy an antistatic soldering station as you can just use an antistatic wrist strap. It is important to have a grounded soldering iron though.
Flux Remover Spray
If you are not using no-clean flux solder then you must clean the flux from the board. This is necessary because some kind of fluxes are slightly corrosive. It also is important for cosmetic reasons if you are fixing someone elses board, when the flux is cleaned the job looks more professional. You can use a flux remover spray for this. I use Flux Off from Cramolin. Alternatively you can use isoprpyl alcohol and a toothbrush.
When you have finished the soldering job there may be some debris on the board like bits of clipped leads or solder flakes. It is a good idea to use some canned air to make sure they dont remain and cause a short.