Without pictures, it's hard to say, however you can dismantle everything from the sink drain down to the U-bend.

If it's a slow drain, Mr muscle usually does the trick. Do double check the labels though, they have a little check list.

If that fails, could be a blockage further down, past the bend, and you may want to get a plumber in. The can either snake it, or use industrial drain cleaners.

This stuff is brilliant. https://www.screwfix.com/p/zep-liquid-heat-drain-unblocker-1ltr/166kh?ref=SFAppShare

Thumb over the overflow and plunge

Check the external outlet pipe where it meets the drain. I had a similar problem that turned out to be leaves in the external drainpipe.

The main thing to do once you’ve unblocked it this time, is be religious about not putting fats, oils etc. down the plug hole.

I suspect there's insufficient fall on the section that runs under the floorboards to the soil stack. Over time it's getting blocked. You might need to remove the under floor section, clean it out then refit with a bit of a fall

Those push fit waste fittings scream Bertie Bodgit. Only appeal to the most incompetent diy-er. Good luck with the rest of the house.

UK plumber here. First, put a load of washing up liquid down the plughole. Then follow that up with boiling water, maybe a few kettles worth if needs be. You can’t do any harm by trying that as it is draining, and you don’t have to take anything apart and risk not getting it back together.

If that alone doesn’t do it, get yourself a decent plunger. Monument’s Master Plunger is fantastic, you can get it easily online or at Screwfix. Block up the overflow (if there is one) with wet kitchen roll or whatever you’ve got to hand. Then start the water and let it build up. Then, holding the blocker on the overflow in place with one hand, push the plunger down and then pull it back up again. You will feel pretty strong resistance if you’re doing it correctly. Pump it like that a few times and then release and see if it’s cleared.

If that doesn’t work, you may have home emergency cover with your home insurance or British Gas, Homeserve etc. Most people do have it in some form or other and don’t realise it.

That’s about as much as I’d recommend for someone who doesn’t have any experience with these things. I’m sure you’ll do fine! Good luck!

Undo the plug and waste and see if the plug and waste outlets line up Properly

It doesn’t sound like you have a lot of experience or tools. Try the upvoted plastic snake mentioned in the most upvotes comment first. You can get them at most hardware stores or Amazon (in the US), they are easy to use, and solve almost all most drain clog problems. If the plastic snake doesn’t fix the problem then move on to next steps , like using a SINK plunger (not a toilet plunger)or removing the drain trap. Depending on your plumbing, removing the trap and getting it reinstalled can become more than a simple job. You’ll get a sense of difficulty when you get into it. Good luck!

It depends on your setup; you might need a large wrench.

Here's a YouTube vid showing someone open up a trap to clear a clog. Maybe it will help you:

https://www.youtube.com/watch?v=8PI0iaIB6-4

What’s under your sink? Is it plastic or metal? If it’s plastic, you might be able to loosen the nuts with your hands. If metal, you’ll probably need a wrench. That’s it for tools.

Loosen the nut on either side of the trap (the bendy part that dips down). Once they’re both loosened, it’ll come right off. Be ready with the bucket, there’s about a cup of water in it.

Once you’ve fixed the clog, replace it by reversing the steps you did to remove it. If it’s plastic, once you’ve tightened it, run hot water down the drain for a minute. That’ll soften the plastic enough to get another quarter turn on the nuts.

You can run a snake or use a plunger without removing the water. If you're looking to use something like Drano, don't.

Fill it up with more water then plunge- I work as a custodian and unclog sinks all day long.

If it's really bad- shop vac the water from the drain then either snake it or clean out the p trap

Put a bucket under the trap underneath, and disconnect the trap. That will get rid of the water. It might also solve your clog issue, and if it doesn't, it will be easier to snake the line, once the trap is disconnected.

you dont have to remove standing water. get a bucket. put it under trap. disassemble trap. drain water into bucket. clean trap. reassemble. test.

Enzymatic drain cleaners (Green Gobbler, etc) can be used as a preventative measure.

The danger of clogs can really depend on what kind of piping you have in your house, too. Old cast iron and galvinized pipe is all rusting from the inside, so those flakes catch things quite easily. If you have PVC, or ABS, liklihood of a clog is much lower if you're not flushing something you shouldn't.

If you have old metal pipes, using Green Gobbler on any of the smaller drains would be helpful. With plastic pipes, it's usually just the kitchen that gets clogged with grease.

A good preventative maintenance routine is to close the stopper or strainer, fill the sink with hot water, and then open the drain and let it flow.

Put a little baking soda in it then pour vinegar down it. It will foan up and might open it up.

Hot vinegar and baking soda add the baking soda last and cover the overflow with a rubber sheet

Pipecleaner

AIRSNAKE

I’ve used a plunger in the drain and it unclogs the overflow. If the plunger will fit on the overflow it might work also.

Get a long drain “snake”. It’s a long piece of plastic that looks like a zip tie (it’s a couple of bucks at most). You stick it into the drain, when you pull it out, the hair gets stuck in the little notches. Be warned: it may be the most disgusting thing you do in your life. Wear gloves. But it will be magical.

Did you get the kind that dissolves hair and use really hot water to flush it? Sometimes it takes more than one bottle. And you might need a little drain snake which is sold with the bottles sometimes. Also if your drain doesn't have any sort of guard over it are you sure something didn't fall down the drain and get stuck? That happened to me with the top from a shaving cream can.

I straight up used a plunger on the drain, did it a few times and fixed the problem

If your problem is long hair, the only thing that’s ever worked for me are these plastic snakes with serrated teeth. $2-5 at a hardware store, and you can reuse them forever.

Drain snake. You always want to use something like a snake or a plunger before resorting to chemicals, because if the chemicals back up and don't drain, you have a caustic mess on your hands.

BIoClean xand Endure are good biological treatments for drains. Use about 4 ounces in each drain monthly to create microbes that clean away the soap scum and greases that build up to clogs. A cup of distilled vinegar each month also helps in removing sludge and scum build-up in the pipes.

I regularly (monthly) pour a huge pot of salted, boiling water down each of our drains. It seems to keep things flowing.

In theory yes, they’ll inevitably happen over time. However that timeframe can vary drastically. Kitchen/laundry drains take the most abuse. A properly plumbed, self-scouring drain should clean small, naturally occurring blockages themselves. However adding all these fats, greases, laundry soap scum, etc in the equation, it is inevitable that there will be some build up over time. Especially with an older material like cast iron.

However, this can be nearly prevented entirely with recurring drain treatments using an enzyme based bacterial cleaner (NOT DRAINO) I always tout BioClean as one of the best examples. Hope this helps a fellow drainiac!

In all the years I've been replacing sewers, I've never had to go back and unstop one. (I'll knock on wood now)

I would spend my time doing something else and not worry about it

So at initial inspection, your bath drain has a good drop to it. This lets the water flow. It doesn’t look like the sink basin or after the fitting has the best flow. It’s not good. But it’ll survive.

The T fitting looks like a bit of a problem. You can get a fitting that promotes flow going into one direction.

Easy fix: What I would do is cut out the T section more than a few inches away from the fitting, all three sides. Then you can inspect and clean. From here. I would get three straight compression couplers to make it way easier in the future. You could put the same section back in. Or you no n get the fitting I was talking about earlier on and a bit of pipe to promote a bit better flow. You can make that solvent weld fitting. But if you want. You can also get the compression fitting for that.

You could also investigate to see if the pipe on the back wall has a good flow and see if you can improve it by adjusting the pipe under the basin and probably cutting an inch out.

If you can get under there to mark it out, try this tee from Screwfix

It's a universal fitting with a sweep on it to move the water away from the tee piece.

The one you have, the water from the bath is just hitting that tee and backing up on itself.

I'm guessing your pipework is 40mm which is why I linked that tee but you might need a 32mm depending on what your pipework is.