I have been struggling for almost a year before I found out how to use the RA and DEC setting circles of my telescope. I do not have a sophisticated GOTO on my scope: I am the GOTO myself, and that should be (HAS to be) enough. First of all: we do need a reliable star atlas.
You can use a software package like Skymap (they have a very nice demo version with unlimited usage) or a printed atlas, such as The Cambridge Star Atlas by Wil Tirion.
Both give per celestial object the Right Ascension (RA) and Declination (DEC).
The problem is how to relate these RA and DEC values with the setting circles of your telescope.
On my particular telescope the RA was incorrectly labelled, and that caused me a LOT of trouble.

Here is how to check that yours is correctly labelled (that is if you live on the Northern Hemisphere; I assume for the Southern Hemisphere everything is just reversed).
As I move the scope from the Eastern sky through the zenith towards the West, as the celestial Sphere appears to rotate, the numbers on my RA circle get SMALLER.

If this is not the case then you need to renumber the RA circle to make it work properly. I used self-adhesive stickers with the numbers 1 through 24. Starting at 0 (leave it 0), change 23 to 1, 22 to 2, etc, etc. 12 will remain the same.

Then, and especially with a correct polar alignment, you put the scope on a known object (star) and then rotate the setting circles - as accurately as you possibly can - to the coordinates of the star: both the RA and DEC.
To refine the aim at the 'known' star I currently use my webcam and K3CCDTools to make sure my starting position is as accurate as possible. I use the reticle crosshairs of K3CCDTools and when the star is right in the center, I set my setting circles, while keeping one eye at the laptop screen.
Hint: use a magnifying glass to set the circles very exact.

Please note that the RA is divided in hours and per hour in steps of 10 minutes.

The following example explains it all.
Let us assume the know star is Regulus and we use a star map to look up the RAand DEC of Regulus: RA 10:08 ; DEC 11 degrees 58 minutes (see images) .

The RA circle is set at 10:08, this being the RA for Regulus.

The DEC circle is set at 11 degrees 58 minutes, this being the DEC for Regulus.

Now you can move the scope in RA and DEC to the position of the object you want and that should be in the field of view of a wide to medium field eyepiece.
Keep in mind that since the RA circle is not driven you will have to keep recalibrating it to the correct RA as long as the drive is running. I just set it at the RA the current object in the eyepiece before continuing to the next object.
You will find out that you do not need to reset the DEC: once correctly set it remains that way.

And to be able to read the setting circles I have made a little lamp to shine on them.
I bought a small lampholder, a halogen 12V 5W lamp and a small switch (for in the power cord).
I mounted the lampholder INSIDE the cap of a film canister, and cut a small window in the side AND the bottom of the canister and put the cap with lampholder and lamp on the canister.
I used cable binders to fasten it at my scope and when I switch it on the light falls on my RA and my DEC circles!!!
Of course I only switch it on for a short period of time, so it will not get hot.
Maybe I have to add a red cover or I might buy a red halogen lamp, I will see.
But it works very nice in daylight with the roof of my observatory closed!

And there is more: Advanced usage of the setting circles with WEBCAM assistance.
Hope all this helps.