On the electronics and effects front - in terms of things like loopers, or recording, etc - there are a lot of very cheap and highly functional digital multi-effects units on the market these days, which would be a good place to start. They will typically include looping and just about any other effect you could think of. Many of them also have the proper interface to plug in to a mic-in port on a PC if you want to record or do things on your computer. These units aren't awesome quality, and people who are doing "real" recording or live performances may scoff at them in general, but when you are just getting started, don't know what you want, and need to experiment in order to learn, it's a really great place to start. Look at places like guitar center, sweetwater, etc and look for a category of "guitar multi effects pedal." Brands like Zoom, Boss, Line 6, etc. If you look for used gear on guitar classifieds forums you can probably get something nice in the $100 - $200 range.
On the pickups front - it really depends on what sound you're looking for. An acoustic ukulele with a piezo pickup (usually installed as a strip under the saddle) will sound very much like an acoustic ukulele, just louder. Most are active, meaning there is a battery and a small preamp onboard. Some are passive - the pickup connects directly to the jack. Generally, active piezo pickups will give you a more even tone response. Some units just require the pickup and a jack mounted in place of your tail strap button, other units need a hole cut in the side for a control panel that lets you adjust the tone. If you want the "acoustic" sound, you can also skip the pickup entirely and just use a microphone. A solid body uke with a typical magnetic pickup will have a tone a little closer to an electric guitar (although by virtue of the scale length and string selection it will still retain some of the uke-like tone). It's worth noting that such instruments, with magnetic pickups, also need steel strings in order for the pickups to work - so the feel of playing one will be different than a traditional uke with nylon strings. The choice really just depends on what tone you want and what kind of instrument you want to play - everything "downstream" in terms of effects, amps, and so on is essentially interchangable.
In terms of amps, again it really depends on what you want. Generally there are two or three classes of amps. Amps aimed at electric guitars have a lot of tone-shaping built in, basically to simulate the traditional expected electric guitar tone. If you want your uke to sound like an acoustic uke, an electric guitar amp would be a bad choice, since it'll roll off much of the characteristic frequency range. Amps aimed at acoustic guitars or basses are generally much flatter in terms of their tone response and preserve the sound of the instrument more accurately. If you want an electric guitar-like tone, get an electric guitar amp. Otherwise, or if you're not sure, try to stick with amps marketed at acoustic guitars or basses. Some modern electric guitar amps have typical electric guitar effects modeling built in to them (usually the basics like reverb) and most will have an overdrive channel meant to simulate a distorted tube amp tone. Worth considering if you want to go that route.
