Cables - any 1/4 inch mono instrumental cable from a reputable company will work just fine.
Most local music store varieties will work just fine.
Avoid no-name ones that are really cheap on Ebay, etc, as they might cause unwanted hum or not be very durable.
Not much point buying very expensive cables - you won't notice the difference (unless perhaps you are using amplification gear worth tens of thousands of dollars).
Amp - depends on your amplification needs.
Do you just want something small to play in the house?
There's plenty of small amps for home use such as Roland Microcube, Mobile Cube, Yamaha THR5 or THR10, various VOX amps.
They often give you the option of running on batteries or mains power, which is convenient if you want to take your amp to play outdoors.
Electric guitar amps tend to give you the most options as many have built in effects and you can experiment with distortion and overdrive.
Most electric guitar amps have an "acoustic" mode too, so will more-or-less be like an acoustic amp anyway.
Acoustic guitar amps supposedly model acoustic sounds better if you want the cleanest and most natural sound, but bear in mind they won't give you options to experiment with distortions and many also don't have built-in effects.
Need something more powerful to perform with at gigs or play in a band?
You'll need something larger and more powerful, at a minimum rated at 45 Watts.
Lastly, you should use guitar amps. From the amp's perspective, ukuleles are just guitars with less sustain and tuned a bit higher.
If your husband's guitar amp works fine, no problems getting the same one or continuing to share it (many often have more than 1 inputs).
