TL;DR: Pearl Exports and Sabian B8 cymbals. Seriously consider new drum heads as an initial upgrade.
The full version: Drum kit is not so different from cycling. Like the bike, the drums are just the start. There are three categories of equipment to consider with drumming: drums, cymbals, and hardware/stands/accessories. One of the better things to do is figure out your budget. $1000 or so will get a solid setup that will generally not need substantial upgrades.
As for the drums, Pearl Export kits are well regarded solid beginner kits. Look for a four or five piece kit, including a snare drum, kick drum, floor tom, and one or two high toms. If you buy new (and many used sets too), you’ll get adequate hardware and entry-level cymbals, too. These are acceptable for beginners. There are plenty of other great choices from brands like Gretsch, Tama, Yamaha, Mapex, and PDP. If you buy new, you’re looking at around $700 for an entry level kit comparable to the Pearl Exports with hardware and brass cymbals. More money typically gets you higher quality drum shells and more robust hardware. If you buy a new kit online, read carefully as to what is included. Some drum kits are just the drum shells, others are full sets with cymbals and stands.
As for cymbals, you will want to start with a set of hi hats (2 cymbals, one top and one bottom), a ride cymbal (mandatory for jazz), and a crash. There is not much love amongst drummers for the lowest level brass cymbals typically speced with entry level kits. You’ll get more quality for your money with budget bronze cymbals like Sabian b8 cymbals, menial MCS, or zildjian ZBT/ZXT. There are literally tons of these Sabians and Zildjians on the used market as drummers sell these to fund more expensive upgraded cymbals.
As for hardware, look for quality screws, clamps, and fasteners and stable legs. Double braced is not required, but the extra weight will add stability. Gibraltar, PDP, Pearl and Tama are solid brands with options at each price point. You will want a straight or boom stand for each cymbal, a dedicated hi-hat stand, a snare drum stand, and a kick pedal.
As for upgrades, it’s a good idea to leave $150 or so to replace the drum heads, even if you buy a new kit. They’re arguably the cheapest and easiest way to improve your drum set. With daily use, you can expect 6 months of use from a set of heads. See your local shop for assistance here.