You don't NEED anything to train, but there are some tools that make the experience more enjoyable, help you make better progress or break plateaus through structural support, or help technique when training. The following is a list of what's in my gym bag and why it's in there. If you're just getting serious about training, use this as a guide for what to purchase.
1) Lifting belt
A good quality belt is indispensable. The belt is NOT for protecting your lower back. It's to provide stability to your trunk which will help you lift more weight than you would without it. Therefore, the belt will help you get stronger. I sometimes hear guys say that they don't use a belt because they want to "be able to" lift without it. That doesn't make sense to me. Training properly with a belt will help push you past a plateau and make you stronger than you could have been without ever using one.
I use a two-prong 13mm Inzer belt.
Sometimes, we want to load an intended movement with more weight or reps than would be possible just using our own grip. Grip training is important, and straps should be used for specific purposes and not for every pulling exercise, but it's also the weak point in the chain between, let's say, your traps and the barbell. We can load our traps heavier with straps than we could without them. No brainer.
EliteFTS makes some great quality straps.
You need to be able to grip the bar. Chalk is a necessity. If your gym doesn't allow it, they're cringe.
You can buy some good stuff at EliteFTS.
4) Flat Shoes
Pretty much all of my footwear is minimalist, but it's especially important in lifting. You need as much contact with the ground as possible. After all. that's where the power transfers when you're jumping, squatting, deadlifting, shoulder pressing etc. So a good, flat shoe with the least amount of material as possible between you and the earth is a necessity. Stop lifting in marshmallow-like running shoes. You're unstable in them and if you were to take a slow motion video of you jumping in them, you'd see what I mean.
I mostly lift in chucks and just buy a new pair when they wear out.
This definitely isn't anywhere near a necessity, but the Slingshot is an awesome tool that makes a big difference in your progress. Just like the straps or the belt, the Slingshot will allow you to use heavier training loads than you can without it, allowing you to overload certain portions of the lift that would have been impossible before. For instance in the bench press, the Slingshot will assist you at the bottom of the movement more than the top, allowing you to use more weight on your triceps than you could have without it.
You can get it straight from the manufacturer, right there.
4) Knee sleeves
Even if you don't have knee pain, sleeves will help your proprioception and movement patterning, which will in turn give you better results by making your technique more efficient. "Knowing" where your knees are is why I like sleeves. The warmth and a bit of support feels great too, and I can definitely push harder in sleeves than without them.
I prefer a heavier, sturdier sleeve like these ones.