If the 700R4 won't shift out of first gear, it's most likely the governor. The governor is under the tin cover on the LH side, towards the rear of the case. You can pull the cover with large channel locks. I use a pair of dust cover pliers that remove the cover without damaging it. With the cover off, the governor pulls straight out. Look for the plastic gear to be stripped. If the gov gear is okay, check the valve for sticking. If the governor checks out, you will need to remove the pan and valve body to check for a stuck valve or valves. The first thing I would check would be the 1-2 shift valve train, next would be the modulated up shift valve.
If you don't understand what these things are, you probably aren't ready to dig into the 700R4. It is a fairly complicated unit. I will tell you that there is great information available for this transmission. The ATSG rebuild manual is a must have if you are going to disassemble the unit. The GM "700R4 Principles of Operation 2nd edition" circa 1983, is a very good book to have and read thoroughly before you think about repairing this transmission. This book was printed by GM for their factory techs and provides a very detailed description of the unit's design and operation. Just a hint for those serious about learning this transmission, this book is out there for free download if you do some searching. Try a Google search using the name as stated in quotation marks above.