I would choose a special spot for him to pee/poop - a corner of the yard or a section on the street, whatever. Anyplace with not too much distraction, traffic, kids playing, etc. That means when you take him out of the crate, he goes right to the door. Same door, every time. Have small training treats in your pocket - not full biscuits. Make him "sit" at the door but be careful not to squeeze his hind section too much to make that happen, lest he pee right away. As soon as he sits or even gets really close to the door, give him one treat with a verbal approval. You want him to get in the habit of going to the same door every time when he has to go out. Take him to the "potty" area. You'll have to linger in that spot for a bit, and watch him like a hawk so that the second he pees, you reward him verbally and with a small treat. Same thing with poop - that's harder to predict, I know, but time your outside trips to meal times.
When he comes back inside, go into the kitchen or other semi-contained area but not the crate. That's playtime. Let him have a few minutes to just wander around before you get into hardcore play - you don't want him to pull you back in from outside because he expects playtime, you know?
You want a family pet so you don't, in my mind, want a dog in the crate all the time. You want him there when you go out, during the night, and when he needs to get away from chaotic play with a bunch of kids visiting or other things (dinner party or Thanksgiving prep, when he's underfoot). The crate isn't punishment - it's his special den. But for now, no playing in the house at large where there are carpets or things that can be chewed. If you can't close off your kitchen or back hallway, get one of those open pens (like a toddler play area) that can be formed into the shape you want.
Be sure you don't have him wearing any kind of collar/harness that can get caught in the crate bars, or chewing on anything that's a choking hazard in there (stuffed toys, for example). Google this or talk to the vet so you are well informed. Just because it's marked "doggy toy" or "collar" doesn't mean it's safe. He should have water but usually no food - you don't want him filling up and having to pee/poop in there. If the crate is large, consider walling off half of it while he's still little so he doesn't have room to pee/poop in there. When he gets bigger, remove the partition.
I agree with JB that you'll be cutting back the number of feedings and naps as he gets older.
The time you spend now will be well worth it. If he's only 3 months old, he's not going to be reliably crate trained anyway, regardless of what the prior owners did. You're lucky that he's young - we rescued a twice-abandoned (and abused) 1.5 year old dog, and she still has crate and potty issues! But she's come a long way.