I have a drain that runs slowly, so if, for example, I shave witht he hot water running, the basin fills. If I then shut off the water, it drains pretty slowly until it gets down a ways, then it “burps” and a few bubbles come up, and it flows the rest of the way more quickly. This behavior continues repeatedly, so it’s not like the burp indicates something has cleared and all is well.
What could be the problem?
Sounds like a generic “slow drain”, probably got a bunch of crud built up in it over the years.
Remove the stopper and see if there’s a visible amount of ick in that first few inches of the downspout, you might be able to swab it out easily (wear a rubber glove, it can be nasty).
Put a bucket or basin under the trap, get an old towel or two, under the trap and flush it clean (I find a good spray from the garden hose will blast the scunge out nicely).
Take a look at the drain where it goes back into the wall (the horizontal bit). Swab out any visible crap.
Reassemble and run some water down, if it’s clear then you’re all set, otherwise the problem is further down the line.
Remove the trap again and get yourself a pipe snake (you buy a 25 foot snake for about $20 at the hardware store). Follow the directions for feeding it into the pipe and cleaning out whatever hair, soap, dirt, skin and other accumulated gunk has built up in there. Nothing beats mechanical action for this.
Reassemble and run water again. You can try some of the other remedies like pouring some baking soda into the drain and then vinegar (foams like mad and may help clear things up), also pour a kettle of boiling water down (be careful handling, or course). I don’t recommend drain cleaners (Drano and their ilk), they never seem to work for me, I’m told they can make things worse (turn a slow drain into a stopped drain) and then you’re left with a pipe full of dangerous caustic chemicals for someone else to clean up.
Drano isn’t all that caustic or dangerous, but is inefficient in clearing slow drains because most of it runs through before acting. In fully clogged drains, there’s often too much water diluting the Drano to make it very effective. But it doesn’t hurt to try it for simple drain clogs. Plumbers use much stronger stuff, but prefer to locate and clear the clog mechanically in most cases. Which is a good idea, because the chemicals will be very ineffective in a drain once the flow is re-established. Some of whatever caused the clog (usually hair and soap) will remain, and it will clog up again in short order.
Using the plastic bucket is important if you don’t want to clean up a big mess, but I think you forgot to mention removing the trap in the second step of your instructions. I’d mention that its a good idea to replace old metal traps with plastic ones. They’re inexpensive, don’t corrode, and easier to get apart and put back together for cleaning.
Typo - for “under the trap” read “UNDO the trap”.
If the drain is almost completely blocked (as it sounds like in your case), a plunger can be very effective. Even putting your hands underwater in the basin, with the palm covering the drain hole, and making a plunging up-and-down action can sometimes shift a blockage. I have a slightly fancy plunger called the “Drain Bomb” that has a piston and is capable of delivering a lot of force.
Note that for a plunger to work, it has to be immersed in water.
Also, if your sink drain has the pop-up type drain plug with the lever on the rim of the sink or faucet, it is possible for the mechanism to trap hair and soap suds and restrict the flow, this is usually easy to disassemble, clean and re-assemble from below.
I had one of those recently. Solved it with an enzymatic drain cleaner I bought at Lowe’s. Already posted about it in this thread, so I’ll just leave that link rather than typing it out again.
IANAPlumber or anything similar…
I think that most house plumbing (on the drain side anyways) are vented to outside at some point ‘downstream’ to help with getting air out of pipes without ‘gurgles’ or other problems that trapped air can cause (~airlock). A blocked vent can cause drains to not drain as well as they could, ime. I lived in a rental house (older build) once that had a bathroom sink that did exactly as OP’s is doing, and out of curiosity, I ran a small airhose (fish-tank airline) through pipe for about 10’ or so, and the symptoms disappeared markedly. I didn’t leave the airline there, of course, but it did validate my suspicions that that portion of drain piping wasn’t vented adequately.
This may, or may not, be what is happening, of course. I am just throwing this out as a possibility other than physical blockage (as posited in other posts above). I don’t know how many gallons of drain cleaner my wife tossed down our slow-drain, but the chemicals (and plunging galore!) did zilch for making it drain faster (and without gurgles) - but the airline did make a difference immediately. Just sayin’… 
Google “drain airlock” and you’ll see many entries on this exact thing (if I am understanding things correctly) and how installing a simple ‘vent’ will alleviate drain issues as per the OP. HTH
Burping = block vents.
I don’t think it’s a vent issue, as I am having no problems at all with the toilet or shower drain, buth of which are plumbed into the same vent stack.
I bought a Zip-It/ a couple of years ago and haven’t had to use drain cleaner since. It works great to get hair out of the bathroom sink and the shower.
An alternative to a snake is the Drain King. It will flush out most old clogged pipes quickly and without the disgusting gag-inducing atrocity of a snake, added plus - you don’t have an icky snake to clean and store afterwords.
In an old house you might have to work the pipes from several locations (whether using a snake or Drain King) because the pipes are likely to have buildup in various locations along their way to the sewer. There should be ‘clean-outs’ after the plumbing in most sinks and at a large one where the stack enters the sewer, where you can access the pipes with a snake.
It sounds like no drain vent or a blocked drain vent.
Ahem.
It really doesn’t make sense that it runs fine after the initial trouble then.