First of all, do you have an active or a passive pickup in your ukulele?
If it's active, that is, it has a battery, you won't need a preamp. You may or may not have a volume control, and/or a bass/treble or equalizer.
If it's passive, your amp will need a built in preamp, or you will need an external preamp or DI box. A passive pickup gives out a very low powered signal when plugged in. Which is great for avoiding feedback with a mic. But can be a problem if you want to plug into a PA system. Like at a church or club or gig venue. Many amps have built in preamps. So if you have a passive pickup, that's a good thing.
Power. How much power do you need? Well, where will you be playing with it? Just at home or with a few friends, you don't need much. 15 watts can do nicely. If you think you're going to need to fill a large room at a venue, 50-100 may be needed. The more power, the more money the amp costs. And usually, the bigger and heavier the amp is due to bigger speakers in the box.
If the pickup is for an acoustic ukulele, you will most likely want an acoustic amp. These more closely replicate the sound of an acoustic instrument as opposed to an amp for electric guitars.
Will you be using a microphone as well as an instrument? Then you need an amp that has an XLR mic jack. (With switchable phantom power for condenser mics.) Plus a 1/4" jack for your uke's pickup.
Special effects
There are lots of amps available; some with effects some without. You can always add boxes (foot switches) with effects at a later date.
My amp is a Boss Acoustic Singer. It has effects for vocals that are applied to the microphone signal. Plus it has separate instrument effects that work through the jack receiving the instrument's signal.
Two of the most common effects are reverb & delay/echo. At a minimum the amp should have Bass, Mid & Treble controls in addition to the volume.
I have always been warned away from buying used amps. Mostly because the electronics and the wiring/solder points can go bad. Plus the speaker cones can start to go bad. All of which is difficult to hear or find with a quick test. Get a guarantee or find out the return policy before you buy one.
Get some good quality cords. Cheap cords can fail quickly or leak signal or just don't have enough wire in them to provide a clear signal. The more copper or aluminum wire in it the more expensive it will be. Ditto shielding. TUS did a podcast about high end amps and cables a couple of years ago.
If I remember right, anything over 25 feet will need a DI to boost the signal.
You can also look at wireless transmitters that you attach to your instrument & the amp. I don't know much about those.
This is a starter answer to help point you towards what you want. Enjoy your search.