Greetings,
Tools?
Well, I like things you can easily explain.
Graph paper and a pencil. I like maps...
Flashlight - small LED that fits in pocket - non threatening. DO NOT take the 6 D Cell Maglight. I know someone in highschool that was using one of those and run in with police considered it a deadly weapon.
Paracord - Cheap, easy, strong, and small cord that can be used for many things. Never been questioned for having it. Better to have and not need than need an emergency rope/binding and not have.
Leatherman/Gerber multi-tool. In most areas they meet legal blade length and if you have it in your pocket/leather pouch, let the officer know it is there BEFORE they search you. ONLY give them that information that it is on you if the conversation seems to be going that direction. If they find it without you telling them, you can get in serious trouble though. But it can always be easily explained away as a simple tool if you explain/tell them at the correct time.
Leather gloves - no brainer - explained usually by "I do a lot of walking in the woods. You ever get stuck in thorns/thistles?"
Jeans - Protect your skin
Long sleeve/jacket - protect skin
Good comfortable boots WITH some ankle support - Protect feet and reduce risk of injury to ankles.
Strong leather belt able to hold your weight - Style doesn't matter, utility does. A hardware store belt that can support your weight can mean the difference between that spot that looks out of reach and that amazing new find.
Tic Tacs - biodegradable brightly colored bread crumbs if you are in a larger place. They are also good for marking safe/unsafe areas. Etc. And popping a few in your mouth before the police get there helps...
Cigarette Lighter - Cauterize ends of paracord so it doesn't fray. And you never know when it helps.
Cell phone with mapping/gps/geocaching apps.
Camera - pictures of nature and birds in primary memory, exploration on SD card. So when they look at what you were photographing it is exterior shots, birds, plants, wildlife, etc. Makes it look like you are an aspiring photog.
The only thing above that is in the "grey area" I have found was the leatherman/gerber. But have never had one confiscated. And I have used them to pick locks in the past... A multi-tool is a lot easier to explain than a full set of pliers and wrenches. You carry too many tools and they will assume you were trying to scrap the place.
In the vehicle:
Tools, and rock climbing gear. All IN THE TRUNK.
Also, jerky/dried fruit and liquid INCLUDING PLAIN WATER to wash out cuts/eyes/etc if need be. Iodine for wound care/etc. Small first aide kit. All in passenger compartment.
My feeling is:
Take nothing in that looks like you are a looter/vandal/thief. - reduces likelihood of severe consequences if caught.
Do no harm.
If you need to "break in" and need anything that can't be easily explained, return it to vehicle/stash someplace before you progress. Carrying that entry gear can change a slap on the wrist to heavy fines/court.
Seriously, B&E tools, pliers, bolt cutters, etc turn you from Urban Explorer to criminal REALLY quick. You are trespassing, and that usually can be talked out of. But carrying those things can turn it into other charges fast that do not easily go away.
You can make a decent set of picks from hacksaw blades. Organizations like FOOLS
http://www.bloomingtonfools.org/ are there for helping you learn lock sporting. And seriously, hacksaw blade picks are an easy throw away. But most of the places you go you will not need them. If you find that door that is locked. Come back and do it later when you have a lay of the land and do those locked areas first on your next visit. But again, RARELY will you need to do this if the building is really abandoned.
Just my two cents.
Have fun,
Hugh