I've been meaning to write this up for a while because the number of posts we get from people who've seen something interesting and then have absolutely no useful data to go with it is, frankly, heartbreaking. 'I saw a light, it was moving, I filmed it on my phone' - look, I believe you saw something, but without duration, bearing, estimated altitude, angular size, and some way to rule out the obvious candidates, there's nothing to work with. So here's how I run my setup at home in rural Wiltshire, and approximate costs so you can adapt it to your own situation and budget.
The basics: A decent pair of binoculars (I use a Celestron SkyMaster 15x70, about £65 from Amazon), a tripod adaptor for them (essential - handheld at 15x is useless, another £15), a compass with a clinometer for recording bearings and elevations, a notepad, and a red torch so you're not ruining your night vision every time you write something down. Total outlay under £100. This is your minimum viable kit. Everything else is optional but useful.
If you want to go further: a decent action camera on a wide-angle mount pointed at a fixed patch of sky gives you continuous recorded coverage, which is how you catch things you'd otherwise miss because you were looking somewhere else. I use a GoPro Hero 11 (the older models are fine and you can get a used Hero 9 for about £80) pointed roughly south-southwest because that's the bearing that's given me the most interesting captures here. I also run a simple weather station - temperature, pressure, humidity - because a lot of apparent anomalies are atmospheric optical effects that correlate with specific conditions. Ruling those out is half the job.
Now - and here I invite the inevitable debate - what are we actually looking for and what constitutes a meaningful find? Over to the floor.