Get out the MSDS for the chemicals you are using and KNOW how to handle them. The MSDS will tell you all you need to know and you should be familiar with the proper protection required for any chemicals or reagents you are using. This is particularly true if you spill some on you, on the floor, on equipment, on your labmates and so on.
Since you say you are using a lot of chemicals you probably should have access to nitrile gloves. If you find that you need nitrile (after reading the MSDS) then order some from your vendor.
If your PI refuses to order them you should report him/her. This is illegal and not something to screw around with, seriously.
I'm not trying to be a b*tch but sometimes labs are just too lackadaisical about safety and it's a serious issue that should be handled appropriately.