I’ve talked to a few people who want to get into canyoning, but don’t necessarily want to spend too much money. Very reasonable! Here’s a short equipment list with a link to a cheap item and a more expensive item, depending on your budget and kit buying philosophy. A lot of the cheap stuff is from Decathlon, cos I like their quality/price ratio and they have a good returns policy.
- Wetsuit (one piece or two pieces) at least 5 mm.
- Neoprene hood (if not part of wetsuit)
- Harness 
- Fig 8 or Petzl pirana with twist lock krab like the Petzl Am’D (twist lock version) 
- Personal anchor/Lanyard with two karabiners
- Head light or waterproof headlight (Petzl Tika)
- Rope ascending gear – prussik loops, jammars, whatever you prefer.
- A few spare quickdraws, slings and krabs – whatever you have
- An old rucksack.
- Old or cheap trainers
- Whistle 
- Penknife or climbing knife
 A climbing harness is fine, just add swimming shorts to protect your wetsuit bottom!
 ATC style descenders are NOT recommended. See here for more info.
 For signalling on pitches where no line of sight or verbal communication is possible.
There’s plenty more equipment you could buy, and I’m not suggesting this is all you need to get started (for one thing, what about a rope?!) but if you’re going with someone experienced and well equipped this is probably enough.
Changing and keeping spare clothes dry
At the bottom of the canyon, we pack everything above into our bags, and hike up wearing our canyoning shoes, shorts and a t-shirt (sunglasses and hat optional!). At the top of the canyon we pause for a picnic, then change into our wetsuits, harness etc. The shorts, t-shirts etc go into a dry bag. At the bottom of the canyon, we change back into our hopefully dry clothes (this is where having a towel and a dry pair of underpants is useful!). Then we walk back to the car and change our shoes (sandals or flip flops are nice to have).