In short, no, you don't need lifting accessories to lift, but in many cases they may give a better outcome depending our your goals.
If you do want to invest in your lifting, we've put these in order of what we think is a priority.
What goals need equipment?
Broadly, lifting the most weight possible is done via using and training with a range of sleeves, belts and wraps. What you choose is dependent on if you compete and where you are in your training block.
What would you recommend?
We have a pretty standard answer for this, because we get it a lot. If you can't keep your heels down squatting, getting shoes with heels is a big priority (along with some mobility work). Its dangerous squatting on your toes, and will give you terrible form you'll have to reverse later.
A stable shoe that doesn't shift around is also way easier to produce power in. If you can, try not to squat in running shoes that are squishy, your ankles are moving around in there and the transfer of power is affected.
For most people, heels when benching will allow you to keep your heel down and get better drive off the bench.
If you can keep your heels down, and good belt is almost mandatory for heavy lifting. Its beyond this article, but you get intra-abdominal pressure that supports your spine. A floppy belt creates no pressure and should be replaced.
That increased pressure makes you feel super stable and hugely increases confidence, especially towards max out on squats.
If you are a bigger person with average mobility, we strongly recommend wrist wraps squatting low bar. People that smash their wrists back unsupported for a long time tend to have issues (that's pretty obvious).
In regards to benching, some people like it, some people don't (we've oscillated back and forward ourselves). Its good and bad, but there are lots of options on wraps, which makes it really hard for people to see what they like without spending a lot of money.
We designed our own wraps quite stiff but in multiple sizes. We use the 60cm which is twice around my ape wrists. Some people like a small amount of revolutions, some people love a long wrap to crank more and more pressure.
Although softer wraps sound enticing, you have to crank them to get the same effect, so to us, a stiff wrap is better.
If elect to use wraps, we do recommend wearing them on all warm ups, not just heavy sets, just so your form stays the same.
I lifted for years without socks, and unfortunately, have the scars to prove it. On reflection, its pretty odd, and in 2022, you might get kicked out of a gym for leaving blood all over a bar.
Another personal preference, but quite common in powerlifting for both the support during the lift, and also the compression keeping the joints warm during longer breaks between heavy squats.
We do blocks with and without sleeves here (low bar sleeved high bar no sleeve) but for almost everyone, you lift more in sleeves, especially once you get used to them.
Some people love to have them crazy tight, but most people just prefer very snug. There is something that is just therapeutic about a hard squat session, finishing, then sitting on a bench and peeling your sleeve down. It always feels amazing.
If you teeth grind, and mouthguard should be at the top of your list. If your elbows hurt from low bar and multiple bench, elbow sleeves do offer support.