What forms do I need to file to probate my mother's will?

A lawyer is generally not required for probate administration, but is often beneficial. You seem quite certain there will be no difficulties, and if you are correct, then administration should be relatively smooth. On the other hand, even reasonable people can differ in their opinions of how something should be handled. If something is done improperly, it is possible you could have personal liability as the executor.
Oklahoma provides small estate proceedings for certain estates under $150,000 in value. A detailed description of the process can be found at the following link:
http://smallestates.uslegal.com/affidavits-and-summary-administration-laws/oklahoma-small-estates-law/
You might also try contacting the probate court in the county where your mother lived and ask them what the procedure is. They are unable to provide legal advice, but they may be able to guide you as to the forms you need, and how they can be submitted.
If you are uncertain as to how to proceed, I would suggest you contact a probate attorney. It is usually worth the cost to make sure that everything is properly handled. It may well be much less expensive than you expect. Many estates can be probated for less than $2,000.
Best of luck to you!
James Frederick