I waited a couple weeks to give Lady her first bath, like you I took her home at 8 weeks. It's best to not wait too long because like you said it's good to start getting him used to it.
As for nails, I found puppy nails got very pointy and scratchy - similar to cat claws! That was the point when I decided they needed to be trimmed. I used to do them myself when she was super young - I did them when she was sleeping and usually did a few at a time every couple weeks just taking little bits off at a time. Once she got older though she wouldn't let me do them anymore...probably because she can sense my anxiety! Now I bring both girls to the groomers for a nail trim and pad shave every 4-6 weeks.
Lady (1 year old tricolour) & Gracie (3 year old blenheim)
"Happiness is a warm puppy" - Charles M. Schulz