Don't use wood glue. And generally, I'm not a fan of putting sticky things on records. Take vinyl care advice on the internet with a grain of salt, especially if it seems overly opinionated about what will or won't TOTALLY DESTROY YOUR RECORDS. It's PVC plastic. If you're attempting to be gentle with them, you're not going to destroy your records.

I have a Nitty Gritty machine that I picked up for a song from a record shop that went out of business. These types of machines are the best for cleaning records, but aren't exactly budget-friendly. If you can get one, go for it.

I have buddies who like the Spin Clean-type systems. Much more affordable, you just need a bit of table space for washing and drying your records without messing them up.

For surface-level cleaning, you're on the right path with that brush. I find wet brush methods are more satisfying than dry brushes, which tend to just push dust around the disc and charge them up with static. They're fine in a pinch for removing hairs and things like that, but I'm not a big fan.

I like to use a Zerostat gun (ebay) to knock down the static charge, which also makes it easier to remove debris that otherwise sticks to the disc like a sock out of the dryer. For small surface-level debris like an individual hair or cloth fibers, I use this rocket air blaster which is powerful enough to remove that stuff, especially if you've mitigated static charge.

Before each play I brush my stylus with a little bit of Last Stylus Cleaner. Not only does it keep your stylus in good shape, it helps it glide through the grooves. I like to pretend that it's cleaning the grooves a bit too, but that is almost certainly just a fantasy that I enjoy. It seems pricy for the size, but a bottle will last you a very long time. What it's not great at is removing hairs/fuzz from your needle, for that you should probably use something like the gel stylus cleaners, of which there are dozens and they're all basically the same.

TL:DR - if you don't have access to a vacuum-type record cleaner, you're not really going to get a deep clean. But don't fret, you can have a nice regimen of surface cleaning tools that will help you out.