Slow bathtub drain: how to clean the pipe myself?

The quickest solution is to use a plunger if you haven't already tried that. Keep in mind that if the pipe is really clogged, baking soda and vinegar (which others have mentioned) work well but require some patience - let it sit for a few hours.

If the plunger isn't enough, I'd pour boiling or nearly boiling water down, then pour in the baking soda and immediately follow with hot vinegar. Cover it with a cloth or a plate to contain the reaction and create pressure. Let it sit for at least 3-4 hours, preferably overnight. After that rinse with very hot water. A lot of the time the blockage is made of hair and grease that solidify there, and that chemical reaction breaks down the mess pretty well without needing to poke around inside.

If nothing comes out, you can try unscrewing the trap manually (put a bucket underneath first, it's messy), and clean that and the pipe section by hand. It's a pain but at least you can see directly what's blocking it. If all that doesn't work, then you really do need a serious mechanical drain snake, or call someone who has the right equipment.

Have you noticed strange noises when the water drains, or has the water stopped completely?

I had a bathtub years ago that emptied in slow motion, and before trying anything chemical I did a physical check of the pipe. Often the problem isn't the main drain but the area right under the tub stopper, where stuff accumulates (hair, soap, dust). Remove the stopper and see if you spot any visible blockages. If it's all clogged with hair, sometimes you just need to pull it out with your fingers or a small pair of tweezers and you're done. A lot of people don't think to look there first.

If you don't see anything obvious or the pipe is blocked deeper down, baking soda and vinegar really do work (like others have said), but honestly I'd add one more step: after you've poured in the baking soda and vinegar and covered the drain for about twenty minutes, try pouring very hot water (almost boiling) down it. This helps especially if the clog is made of grease or soap residue. If that doesn't work either, then it's worth getting one of those spiral drain cleaners (they don't cost much) and trying to break the obstruction mechanically. A plunger is less effective on a bathtub in my opinion - it works better on a sink.

One thing nobody mentions is that if you use vinegar and baking soda, the pipe might not be completely clear even if water starts draining again - maybe the clog is deeper down and you're just pushing the gunk from one side to the other. What worked for me was a combination: first baking soda and vinegar to soften the limescale buildup, then after an hour a good flush with boiling water, and only if that didn't work did I grab a plunger to apply real pressure. Oh, and if the pipe is really old, keep in mind that overly aggressive plunging can sometimes loosen the joints - better to use firm but controlled movements than to go crazy with it.

If none of what they suggested works, often the problem is deeper in the main drain pipe, not in the tub itself. Before calling a plumber, try unscrewing the trap under the tub (that U-shaped one) and check if there's a buildup of hair and debris there - that's where it gets blocked most often and cleaning it by hand is really simple and free. If the trap is clean but the water's still draining slowly, then yeah, it's probably worth having a professional take a look because it could be clogged further back in the stack.

When the pipe isn't draining well, before jumping to chemical solutions it's worth trying a flexible plastic rod - the kind that's cheap at the hardware store and lets you reach the clog without taking anything apart. In my case it was able to break up a ball of hair and debris that vinegar alone would never have gotten rid of. If that doesn't work either, then yeah, baking soda and vinegar makes sense, but maybe after you've already tried to pull out the bulk of it physically - that way the two react directly on whatever's left instead of just sitting there doing basically nothing like someone said before.