They do get some advantages, yes.
For one, Bulbasaur has access to Sleep Powder, which will be really usefull in boss fights. He also gets Leech Seed for chip damage and recovery. Probably the best thing about Bulbasaur (not all Grass-types), though, is his IQ tree. I'll list off some of the best ones.
PP Saver – There is a chance for Bulbasaur to use an attack without losing PP.
Sharp Shooter – Raises critical hit ratio.
Multitalent – Adds an extra 5 PP onto each of the Pokémon's moves.
Exp. Elite – Gaisn for Exp. Points.
All-Terrain Hiker – Allows Bulbasaur to walk on lava, water and gaps (like in Spatial Rift). This solves your problem of not being able to travel where your partner does.
Trap Buster – You will sometimes destroy traps by stepping on them.
Stair Sensor – When you first enter a floor, an arrow will appear pointing in the general direction of the stairs. I had this on my Meowth and it helped so much.
So, in conclusion, Grass-types do not have as many benefits as Water- or Fire-types, but Bulbasaur is a great starter to stick with since he gets All-Terrain Hiker and other useful moves and IQ skills. (I wish I'd started as Bulbasaur!)
Have fun on your adventure!