There are so many ways to do it...
You seem to have a computer (as you post here), so at the basic level all you'd need is a microphone that plugs straight into the mic-in 3.5mm jack socket on most PCs.
The alternatives depend upon your requirements and budget.
A decent stage or recording mic need not cost the earth. Something like the
Shure PG58 is a great start. Alternatively, a condenser such as the
Shure PG81 works well for me. The problem is that you need a way to connect these mics to your PC.
You also need some kind of software for recording and editing.
Audacity is free and not hard to learn to use.
If your PC has a line level input ("line-in") then your options expand greatly. A couple of the configurations I have used:
1) Using an amplifier as the interface to produce line-level signal
- Uke with pickup into channel one of Marshall acoustic combo.
- Vocal mic into channel two
- Adjust relative levels, effects etc on the amp's front panel, then take line-out from amp into line-in on PC.
- Record into Audacity
2) Using a small mixer
- Condenser mic for uke into a mic socket on Alesis Multimix 8 mixer
- Vocal mic into another mic socket on the mixer
- Adjust levels and FX on mixer
- output from mixer to line-in of PC
- Record into Audacity
There are many simpler and more complicated methods, but both of those configurations have worked fine for me.
It really does depend on what you have, what you can spend and the quality of the results you need to achieve.