I want to move my existing thermostat just a few feet away from where it is now. My question regards the new thermostat wire. Is it okay to splice the new wire and the old wire, if it is okay, any suggestion on method of splicing?

It is OK to splice but make sure you do a good job and wrap each connection separately with electrical tape. I prefer soldering but a good twist connection is sufficient.

P.S. I forgot to mention that you need to kill the power to the stat while doing your splice. You are probably dealing with only 24 V but a short can cause a problem.

I would suggest using small, blue, wire nuts and taping each of them.