id say replace the caps first, you should be able to read the markings on them and order suitable replacements on line, you have nothing to loose and its probably the cheapest option to fix it if its the caps at fault.

there are lots of ways to build power a power supply, first thing is to determine, is your current psu switch mode or transformer based unregulated? if its unregulated it will most likely have a big transformer and inside that transformer there will be multiple winding's that output different voltages, the output of that transformer will be ac, so if you need dc you will have to have a rectifier for each winding for each voltage to convert ac to dc, you will also need some smoothing caps in the line to deal with line noise/fluctuations etc. if its a switch mode it will most likely look like a sealed rectangular metal box with ac in and a bunch of screw terminals with few different dc outputs for various voltages. if you have a picture/schematic of the psu ont he machine then it will be easy to know what you have.

if you going to do some soldering i would highly recommend you buy a ts100 or a ts80p soldering iron, its cheap ish and the only soldering iron you will ever need for the rest of your life (make sure it has the p higher power) they are really good irons and if your new to soldering it will make the process of learning 100x easier, the last thing you want is a crappy soldering iron and to not know if the issue is you or the tool.

ts100 (older model but has a bit more power, uses a laptop psu) https://www.amazon.com/UY-CHAN-Progr...07&sr=8-5&th=1
ts80p (newer model little bit less power but can run from usb-c) https://www.amazon.com/UY-CHAN-Progr...07&sr=8-5&th=1

roll of solder https://www.amazon.com/MAIYUM-63-37-...s%2C148&sr=8-5
solder pump & wick to remove old solder https://www.amazon.com/Tabiger-Solde...4269776&sr=8-3

video how to de solder caps, its on a tv but its the same process.