When drain water from one fixture comes back out another fixture, it means your blockage is beyond the point where those drains meet.
You need to get a good look at the topology of your waste plumbing, to get some ideas about where the blockage is. Clothes washer -> bathtub is a common symptom, because the clothes washer drains a lot of water up high (it has a pump) and the tub drain is down low.
Snaking down a drain is really hard, because of all the curves of the P-trap. If you're lucky, you have a clean-out somewhere, like under the house or behind a wall. Otherwise, you can try removing a P-trap to get the snake in, or removing a toilet. Removing a toilet is not very hard, but can be daunting.
We had a similar problem in this house. There were a few clean-outs accessible in the crawlspace, and our landlord used a hand-held snake in them, but didn't have much luck.
They hired a professional plumber. He pulled a toilet out and ran a huge snake down the drain (not one of the handheld units, but a huge, loud, floor-standing model heavy-duty cable). After running all 75' out, he borrowed another snake from his colleague, for 150' of snaking. This is a single-story house, on a 50'-wide lot, so the snake was well in to the street!
He said he pulled out a lot of roots. This is apparently common in older plumbing. They develop a small leak, and nearby plants grow after the nutrient-rich water, and find their way in to the pipe. It took him about 4 hours.
Plumbing is simple. There aren't a lot of rules to follow. Everything works in obvious ways. You don't need a lot of expensive, complicated tools, and if you do need something big, you can rent it. Anyone can fix plumbing.
But it can also be unpleasant. Screw-on connections may be rusted in place - hacksawing is often easier. There's the spectre of old poop and hair. Things may be difficult to reach - in a cabinet behind a sink basin or in a muddy crawlspace full of rat droppings.
If the job takes a few days, and you're unaccustomed to living without plumbing, it can be a trying time. Especially if there's a whole family in the house, and they aren't sympathetic. A pro will get the job done much faster, since they arrive with the right tools, parts, and experience.
There's a lot of instructions out there, on the internet, at the library, at the hardware store. If you decide to get your feet wet (get it? ha ha) then you will be able to find the information you need.
All traps need venting in all cases. Without a vent, the water seal in the trap gets sucked out and sewer gases can enter the room. There's also all sorts of examples where someone thought they were providing venting, but done in such a way that clearing the water seal was still a possibility. Some of these methods even used to be accepted practice, such as S traps. There's still several appropriate ways to achieve proper venting. For example, in island sink situations, a loop or foot vent is acceptable.
It's hard to say why the previous installation worked without venting. For sure, one way or another, the siphon action was disturbed by the entrance of air somehow, either from oversized pipe or leaking fittings. Regardless, where do you go from here? The downward facing ell in wall needs to be replaced with a sanitary tee for starters. The top of the new tee needs to tie into a vent somewhere, without travelling downwards. Is there a vent pipe nearby, such as from a toilet or shower? Or a pipe could perhaps be extended up into the attic, where it should be much easier to make a run to a vent pipe?
If all that sounds like too much, get a mechanical vent such as jberger suggested. They are not accepted by many codes, but they do work. Just be sure it's attached to a long enough pipe so that the valve is above the flood level of the sink. No doubt this all sounds like a real PITA right now. Once it's all done, you can take pride and sleep well knowing you have a properly functioning and completely sanitary sink drainage.
Best Answer
A partial blockage allows some draining, but will backup when the water flows fast. It's not unusual.
You can clean the line with a snake - let the water run the whole time (to wash the dirt away, and keep down the smell), and go back and forth with the snake letting it spin and clean the walls of the pipe.
I also don't see a vent, but it may be hidden.