I have a fender guitar amplifier, and I can't get any low volume out of it. At level 1 on the dial the volume is sooo loud - no gradual increase at all. It's impossible to play softly. What's going on?

It sounds like (pun intended) the volume pot just needs to be replaced. A good pot is worth about $5 and then you just need to solder it in. That tends to be the tricky part if you can't solder. It takes longer to heat up the iron than it does to do the job. I'd assume this happened either after it had sat for quite a while or after a move.
What can happen is the wiper or the resist inside the pot breaksor even the wire to the pot breaks and there goes your volume control. Depending on what model of amp you have you don't want to run it like this for too long. You can't fix the pot but you do need to replace it. I would assume it sounds like it is either full on or doesn't change the volume at all.
This is one of those maintenance things you should check for from time to time. you may notice that the pots get noisy or inconsistent for a while and then they do this kind of thing.
The attenuator will kill the volume but that doesn't actually fix the problem you have. When the amp is running properly you should be able to get quite a low volume. If you need to run it harder so that it gets into the distortion range then a attenuator will drop the volume down to a practice at home non eviction volume ;-]

On any amp I've ever had the volume control will not have the guitar
going through it at all with the volume all the way down.
I would just replace the volume control.

There can be more than one problem.

Check that a MASTER volume control is turned down as well as the inout volume control.

The other problem is your GUITAR may be too "HOT" meaning it has a preamp in it that generates a high level of output that overpowers the inout volume control. You can make an attenuator from two resistors to reduce the signal into the Fender.

The fender is designed to take a non-preamped guitar so it has a lot of gain.

With two resistors and a couple of plugs you can make a guitar cable that has a built-in attenuator... or you can modify an existing cable.

Google "L pad" for a schematic as we don't have anyway on this forum to send pix. The circuit is super simple and takes fvie minutes with a soldering iron to create one.

Here is a site: (use the first one "L")

http://www.uneeda-audio.com/pads/