Good for you for trying this yourself, as caulking is something that every homeowner should do. It’s a job that will have to be done from time to time. It’s not hard, but there’s a bit of an artistic aspect to it. You need to get the pace right in terms of how fast you move the tube for how much is being squeezed out. You’ll get the hang of it after a bit.
You should get a caulking gun and new caulk. As you are a newbie, get a good gun. There are cheapo guns, but those are best left for people with more skill who can work around the gun’s shortcomings. When you look at the guns, you’ll notice some have plungers with smooth shafts and others have plungers with notched shafts. Get the one with smooth shaft. It will stop the flow more instantly when you release the trigger compared to a gun with a notches.
Technically you don’t have to remove all the caulk, but it’s better if you do. If some of the caulk is failing, it might mean the other caulk will be failing soon. You might as well do it all now rather than some now and some later.
Since this is your first time, I would recommend using tape on either side of where you’ll caulk so you get nice, straight lines. Once you’re more skilled you won’t need tape since you will be able to get a good bead by feel. When you use tape make sure you smooth the bead down very thin to the tape’s edge. If the caulk is thick on the tape, you’ll leave a lip when you remove the tape.
I would recommend practicing first. As it takes a bit of time to get the feel, if you can practice for a while you can figure out how much to squeeze out, how fast to move the caulk, how thick to make the bead, how to smooth it out, etc. before doing it for real. You can practice in the shower. Just wipe away the caulk and do another bead until you have the feel for it.
When cutting the caulk tube, make a small hole at first. You can always make it bigger. There’s a foil seal on the top of the tube behind the nozzle. Your caulking gun should have a straight wire near the trigger that you can poke in the nozzle to puncture the foil. Make sure you do that before putting the tube in the gun. If you don’t, the caulk may come out of the back of the tube when you squeeze the trigger.
It may sound like a lot, but after about 10 minutes you’ll have the hang of it. If you don’t like how it’s turning out, just wipe it off and try again. Good luck!