Easy walk from PH to Caesars'. If the weather is hot, just keep walking north inside of PH as far as you can... cut out the doors than back into the doors at Paris....head towards the rear of Paris, and walk north again. The cobble stone corridor (air conditioned) will lead you into Bally's....keep walking north inside Bally's as far as you can (valet or the registration desk), then hit-the-heat and go outside to the walkover bridge towards Bill's (formerly Barbary Coast) and than across the bridge towards Caesars'.
DO NOT walk to the Palms...it's dangerous . You have to cross the entrance to I-15 (main highway). It's a two lane on-ramp and driver's just don't see you (really badly lit at night too), until it's too late. Between the heat (and it's a tough uphill climb across the highway) and the crazy drivers - just DON'T DO IT!
Here is a link to a great page of available shuttles in Vegas:

http://www.lasvegasadvisor.com/transport.cfm

(for some reason I can't get the shuttle page to pop up alone in this link , so just click on the "Free Shuttle" link within this page)

Palms has a free shuttle from Caesars Forum Shops. I haven't tried this shuttle yet, but have done the next two....

If you have a HET (Harrah's/Paris/Bally's, etc) players card, you can get a ride over to Palms quite easily....There is a HET shuttle bus that picks up along the driveway between PH and Paris(and at Harrah's too). It's free (I always tip the driver though) and about 50% of the time I have had them ask to see a players' card. The shuttle drops you off at the Rio...just walk across the street to the Palms. This is a full sized bus, and I've never had problems getting a seat.

Gold Coast and the Orleans also run a free shuttle (again, I always tip) from the south side of Bill's (along Flamingo near the parking garage entrance). Get off at the Gold Coast and walk across the street. Once the driver asked for a room key, but I just told him I was going to the Gold Coast to eat and gamble with friends (with tip $ in hand), no problem.
This is only a 50 passenger shuttle bus. If the line is long, you may have to wait for the next bus.

Have a good time!