People have different ways of doing it, with practice you'll find what works for you.
Soldering irons struggle to deliver heat to exactly the spot where it's needed. Assuming you don't have a very expensive soldering station, one thing to keep in mind is that when the tip makes contact with the joint, it cools off quickly, then has to wait for the heater in the iron to heat it back up again. This is the challenge with heavier joints.
You want the tip on it's own to be able to transmit enough heat into the joint to melt the solder before the tip cools off. Tips with more thermal mass are more effective.
There's a temptation to use a small conical tip, but these do a poor job of conducting heat at that tip. Many people (including me) prefer to use a chisel tip. I have several sizes but use the 2.4mm and 3.2mm chisels on most motherboard caps. I only use smaller sizes if the larger tips don't feel comfortable. I also have a 5.0mm but it feels too bulky for most situations. The general principle is that larger tips have more thermal mass, which helps them to get more heat into a joint.
The quality and variety of tips varies depending what iron you have, of course, but a ~2-3mm range chisel is the most useful tip IMO.
Add 63/37 or 60/40 lead based solder to the joint. This does a couple things - one the flux in the new solder will clean the surface of the joint so it's more receptive to heat. Secondly it increases the contact area and helps conduct more heat into the joint.
On that first point, you can also add flux separately.
Solder suckers and solder braid are controversial. Some people hate or love either of these. It's partly personal preference and the knack that people have learned with their preferred tools, but I think quality of the tools can also be part of it. I can't do anything but the most rudimentary cleanup with braid, but some people swear that they can clear joints with it. I've even seen it on video, but it could have been witchcraft. 😀
I have a large Edsyn Soldapullt that I've been happy with. I bought it basically because it was the most reputed simple solder sucker I knew of, expensive for a solder sucker ($30) but far short of the cost and complication of a powered vacuum sucker.
Most people seem to buy smaller unbranded solder suckers and I'm not sure if they work as well. I would assume the reduced size means they don't generate as much suction, or at least they're obviously not moving as much total air.
At least with the Soldapullt, the tip is designed to withstand heat reasonably well. You can put the sucker right over the joint while the iron is still on it. I haven't had a great experience trying to suck lead-free concrete but leaded solder comes up pretty well.
All that said, I don't think a sucker is that necessary for capacitors. Personally with capacitors I like to melt one joint and gently rock the capacitor up on that side, then alternate to the other side, back and forth until it's free. Solder suckers can also work but with 2 legged devices I think it's usually easier to just rock it out.
For replacement, you might want to clean the holes with a solder sucker (adding solder can actually help with this also), or just use the needle trick (which I think tends to be easier on the harder joints).