For a private GA pilot, when is an endorsement needed for a particular type of aircraft?
There are two aspects here, one what is required legally and then what may be required locally. 

From a legal standpoint (in your case) the basic PPL will allow you to fly single engine, fixed pitch prop, under 200HP, fixed gear aircraft. If you want to go over 200 HP you will need a High Performance check out and log book endorsement. For a constant speed prop/folding gear/movable flap (all 3 are required to require this endorsement) plane you will need a complex endorsement in your log book. You may want to fly something like a Piper Cub which will require a tail wheel endorsement. If you are lucky enough (and have the funding to do so) you may find your self in a small pressurized single (or twin) like the Piper Meridian or TBM 900 these planes will require a high altitude checkout as well. AOPA has a nice summary of the endorsements here. If you want the full FAA FAR regs you can find them here.  

Now as for a club or school that depends on a variety of factors including their insurance and their personal limits. I'll offer some personal experience which seems to be pretty par for the course. The flight school I fly from requires a checkout in type from every new renter before you can fly it even if you have a PPL and the proper endorsements. They don't have an official amount of time but it usually takes about an hour with an instructor. The former flight school I was at had the same deal with the exception of their Saratoga which they also required an instrument rating (along with their checkout).

My buddy's flight club is somewhat similar. They require a check out in their Archer and Dakota to fly solo. They also have a Lance which they require 25 hours time in type and a 5 hour check out in the plane.

As for your log book you should be logging the full type of aircraft for every flight. For example when I fly the Warrior I make sure to note PA-28-161 and the Archer I fly is noted as PA-28-181 when I record hours this way I can note time in type at a detailed level. 

For what its worth with the PA-28 airframe it seems that some places to allow you to fly the lower rated planes if you are checked out in the higher power ones. The same basic frame was built in a 140, 160, 180 and 235HP variant over the years and has little differences aside from useful load and V speeds. For example at the place I go if you are checked out in the Archer first you are allowed to fly the Warrior with out a checkout.  

This will also extend to ownership should that be the path you go down. In that case you insurance company may require a certain amount of hours of time in type before you are allowed to solo your plane. Keeping that in mind you are not actually prevented from buying the plane and will be insured under the stipulation that the first hours in the plane are with an instructor. When a buddy of mine (instrument rated and close to 600 hours at the time) upgraded from his 72 Archer to a 95 Saratoga HP his insurance company mandated he do 25 hours in type with an instructor before taking the plane solo. The first 25 hours he flew his plane were with an instructor (this included transporting the plane from the factory in Vero Beach to his home field). 

You can legally fly both planes once you have your PPL. What someone will let you fly should be pretty well addressed above. Neither the 172 or 152 requires a specific type rating like larger planes do.