We saw a decent improvement by adding another 4Gb of RAM to our early 2009. Adding memory is very easy to do (even for a non techie) and is relatively cheap, so try that first.
The next step is probably to replace the hard drive with a solid state disk (SSD), which will make a massive improvement, but is a nontrivial operation. It's not especially hard to do, and there are various excellent guides (iFixit) with step by step instructions; but it is fiddly, and requires patience, and a careful touch if you aren't used to fooling with ribbon cable connectors and the like. If you can go slow without distractions, and can follow directions exactly, and can stop and think before acting, you won't have any problem at all. Otherwise, either pay for someone to do the installation, or take the new machine route. But definitely try adding memory first, it might make enough difference to get you by.
As far as whether your drive is failing, I suppose you could try one of the SMART reporting utilities such as
https://www.volitans-software.com/apps/smart-utility/
(not a recommendation, I've never tried it, just one of many that popped up via google). IMHO interpreting a SMART report isn't as simple as go/nogo, unless the drive really is on its last legs, and if that's the case you'll likely find out the hard way in a few days...
Regardless of what you decide to do, make sure you have a good backup before doing it.
edited to add: that early 2009 iMac with 8 Gb ram and an SSD runs El Capitan very nicely. Haven't tried Sierra and we're not in a rush to do so.
[doublepost=1478363065][/doublepost]One other thing: shut down all applications, then run Activity Monitor (in the Applications/Utilities folder). Click the CPU tab and let it sit for a few minutes. Look at the bottom of the window where it shows System, User, Idle; System+User percents should not add up to more than 1 or 2% at most. If it's significantly more, something is running in the background eating up resources; probably an add-on or extension of some kind, but could also be malware. The process list section of Activity Monitor can help you find it.
Activity Monitor is a very useful tool; the problem is, that it helps if you know what your activity looks like when things are running fine, so that you can tell the difference! and most people never touch it unless the machine starts running poorly.