This question pops up a lot on UKC and here is my answer as a relatively new climber and a bit of a gear freak. So firstly, just bear in mind I’m still a fairly inexperienced climber. However I’ve taught myself with this lot. Secondly read lots about gear. As a starting place try the following articles:
Article from TradClimbing+ on buying a rack.
Andy Kirkpatricks advice on a climbing rack (read everything else on his site as well!!!)
Another good gear list from Needlesports, more on their website
This would be my list of ‘the bare minimum’ to get up short (up to about 20m) & easy (up to about HS/VS) trad routes safely:
Set of nuts 1-11 (I got DMM Wallnuts)
Set of Hexes 3/4 of the larger ones (I advise a full set of DMM Torque Nuts)
4/5 Quickdraws a mix of 18cm and 25cm
2/3 locking krabs (make sure one is a big HMS style krab)
2 x 120cm slings
1 x 240cm sling
1 x 400cm cordlette (not essential, but I found it made things easier when starting out)
Don’t bother with cams, they’re expensive and it’s harder to tell a good placement and easy to place them wrong. Once you’ve got up the grades a bit more, you’ll have a better idea of their use. Plus all the usual stuff, harness, helmet, belay device, nutkey, prussic loops. As for what actual gear to buy? Well which ever brand/make/model you like the best. Or choose them on weight. Just remember the really small lightweight krabs can be annoying when fitting lots of clove hitches on them. For example DMM Phantoms are rather small and it’s hard to get lots of clove hitches on the. However they are ace in belays as they’re so light you can use them to setup a totally bomber belay. You can use the slingdraws for threads as the two krabs on the slingdraw can be put back to back to be as secure as a locking screwgate.
For quickdraws you’ve got several options, either buy cheap (eg DMM Prowires), or light and expensive (DMM Phantoms), lightish but normal sized (DMM Spectre IIs), lightish and with ‘snag-free’ noses eg DMM Sheilds, or uber strong (DMM Shadow (great for sport climbing)). The snag free noses reduces the rather unlikely possibility that you might snag a bit of gear on the nose of your quickdraw leaving the gate open and the krab very weak.
Here are a couple of good threads:
Which contain some good advice.