I would like to be able to have my trainer look at my entries and my My Fitness Pal site. Is that possible?

I just gave mine my login information. If they become friends with you they should be able to see your food & exercise log.

you can set your diary to either public which means your trainer can view it even if they don't sign up for Myfitnesspal, or you can lock it with a key which also means they can view it without being a member of MFP.

They can view your diary with the nutrient details you select up to 5 optional ones at food > settings
which is the same spot you choose whether the diary is public, or locked with a key or shared with friends only..

What they can't see is any measurements that you've logged in MFP those are "entirely private" unless you choose to put them in your food or exercise notes.