Q: Does my OS have something to do with the system running slow?

It's not the OS, and upgrading will not solve some other underlying problem.

You said this started lately.  Do you recall anything that changed recently?  A software upgrade?  New apps?  Is your hard drive getting full?

Get a backup on this ASAP as this may be the symptom of a failing drive. Not all that reliable, but open Disk Utility, select the drive, the topmost listing and see what it says for SMART Status: Verified or not? But there are other possibilities.

Open Activity Monitor in Applications>Utilities. Go to the CPU tab. How much CPU is idle? If not much, is there any process or application consuming an inordate amount of CPU?

Switch to the System Memory tab and post a screenshot (Shift-Cmd-4) of the bottom only. Use the camera icon above to paste it in here. Like this.

For a (possibly) more accurate test of the health of the drive get the demo of SMART Utility. In any case, you should never use the computer without a reliable backup.

http://www.volitans-software.com/smart_utility.php

1GB RAM is pretty thin by today's standards, and yes as a general statement more RAM will help.

But also check the space usage on your system HD.  If it's more than about 75% utilized you should consider moving out stuff you don't need there.  As the HD fills up, performance decreases; it becomes particularly noticeable aroud 75% of capacity.

Deal with your RAM and hard drive space first.  That may be all you need to do.

And if you don't have a backup, that would be a very good idea even if your hard drive is in perfect health.

Defragging an OS X hard drive is normally unnecessary and can actually create more problems than it solves.

As the HD fills up, performance decreases; it becomes particularly noticeable aroud 75% of capacity.

This is completely misleading. You can ignore the suggestion to leave 75% of your boot disk free. That is  unnecessary. The system needs around 10GB + some room for anticipated expansion for your user data. A ballpark would be to leave around 20-25 GB free. If you leave anything more than that free it will just be wasted space.

No cleaning or defragging is necessary.

You should post the screenshot I asked for, but 1GB Memory is bare bones, even for 10.5.

the suggestion to leave 75% of your boot disk free.

That's not what I said.  I said when the drive gets about 75% full that's about the time that drive-related performance slowdowns begin to become apparent.  I did not suggest leaving 75% of the disk free.  That would be silly.

Once you make a bootable backup/clone, get more RAM, and ensure that you have about 20 GB free disk space, see:

Mac Maintenance Quick Assist,
Mac OS X speed FAQ,
Speeding up Macs,
How to Speed up Macs,
Macintosh OS X Routine Maintenance,
Mac troubleshooting: What to do when your computer is too slow,
Essential Mac Maintenance: Get set up,
Essential Mac Maintenance: Rev up your routines,
Maintaining OS X, 
Five Mac maintenance myths and
Myths of required versus not required maintenance for Mac OS X for information.

My mistake. I obviously meant it the other way around. That setting a minimum of 25% free (or any minimum percentage figure) is misguided and unnecessary.