Onto my thoughts on recapping motherboards.
If you are going to start recapping a motherboard, you should almost definitely change all the caps on the same series, and preferably all capacitors 470uF and larger.
More than once, I've had to go back to a board multiple times to replace further capacitors months after the initial repair because other capacitors on the board have decided to spill their guts. If you want a recap job to actually stick, and not cause issues down the line, do the job in full at the beginning.
Onto how I do it:
First of all, you need a good soldering iron. Something like a KSGER T12 soldering iron or one of the other cheap soldering irons youtube people fawn over, they all seem to be inexpensive and do a good enough job. I use the D12 tip (and D16 too somethings). I also use solder braid (wick). I have a desoldering station, but don't use it for recapping because it can be rough on the boards, to easy to damage nearby traces compared with a soldering iron.
I normally work with the soldering iron set to 400C, but will boost it up to 450C when working on the well heatsinked pads around the CPU vrms. I add some fresh leaded solder to each leg first of all, this lowers the melting temperature, wets the old solder, and gives your soldering iron more direct contact with the solder joint you are trying to remove. I then heat each leg up and walk the old capacitor out of the board, by pulling on it from the other side. Then I will attempt to clear the old solder out using solder wick, attacking from both sides if I have to. I am usually able to clear at least one hole, but I don't waste too much time on this if I can't get the heavily heatsinked pads clear.
Puttng in the new cap, I have cleared both holes, I pretrim the legs down to almost the correct size, and put it in and then solder it down. I then trim the legs to the correct length, and reflow each joint. Cutting the legs will put stress on the solder joint, so reflowing after cutting is just making sure I didn't crack the joint or something while cutting the leg. If I didn't get both holes cleared, I trim down both legs, but make the leg that goes into the cleared hole a little longer to act as a guide. I then add some solder back to the hole I wasn't able to clear, and walk the new cap back into the board. The cut legs down and reflow both solder joints.
I only replace a few capacitors at a time, keeping an eye on the polarity of each capacitor and the marking on the board (be aware that there could be errors in the screen print of the board, check/document capacitor orientation before you replace any) Before powering up the board, I check I have resoldered all the capacitors properly, give it a bit of a clean with isopropyl alcohol, and double check the polarity of the capacitors.