Max has been to the groomer 3x now, the 1st two he just got a trim (potty areas shaved & face & ears trimmed, and anywhere else that was getting unruly); the 3rd time (just this week) he got his first real haircut, which removed the rest of his wirey puppy hair & left him silky (and so much lighter in color). Max is 5 months old, goes to the groomer once a month for the total package & I brush him daily & bathe him weekly, too.
His 1st trip to be groomed was at 3 months, he'd just had his 2nd puppy vax (which included kennel cough), so he was fine to go at that time. I have him groomed at a doggy daycare, so he gets some serious socialization while he's there.
He loves it!
I think a good groomer is going to listen to what you want, in terms of keeping a puppy cut until you're ready to grow out a show coat. That's really up to you...I don't think we'll be growing Max's hair out too much.
As for the greasy hair...not really sure. I know Max isn't fond of the bath yet, so by the time we're ready to rinse, he's ready to be done, so I have to be careful to make sure I get all the shampoo & conditioner rinsed out well.