Use a towel or rag or newspaper cut in strips. Holding an end in each hand making a "u" around the base of the plant. Cross the ends over (sorta like 1/2 of a figure 8) around the plant and twist them to make a stiff "pot handle".

I use this method when transplanting my thorny 'pets' and it works very well.

I found a link that sort of shows you the general idea.

Here is a link that might be useful: transplanting cactus

With Crown of Thorns I put on my leather gloves when working with them. Any damage that may occur to the thorns is minimal. When moving cactus I do as Wanda suggests, usually using an old T-shirt cut into strips.

BBQ tongs if you need to reach in very far ...diana