I suggest to buff up your team before getting the third badge. he uses electric types so be careful when using squirtle.
getting cut from vermilion city south of cerulean
No Underground Path is in Vermilion City. There is one north of the city that leads to Cerulean City.
Diglett's cave is just east of vermilion city
You talk to Officer Jenny in Vermilion City after getting the Thunder Badge, then she'll give it to you.
There is no gate in Vermilion City, unless you mean to get to the gym, in which you have to cut down the tree.
by the dock
not
Vermilion city.
After getting the 2nd badge go south from cerulean city and use the underground path to continue your journey south to reach vermilion city.
you either use cut just below pewter city (you get cut from the captain of the S.S.Anne in vermilion city) or you go through the other end of the cave on the right side edge of vermilion city (the end of the cave in vermilion city is unprotected so you can go strait in).
Vermilion City.
You have to take it with you.