(updated November 15, 2017)
The two cheapest ways to get around Las Vegas are to walk and to catch the various free shuttles. Here is a list of free shuttles.

The Rio    (updated November 15, 2017):
The Rio shuttle is no longer running.

The Palms  :
The Palms offers a free shuttle that runs every 30 minutes  from 11am – 8pm from the Palms main valet  to the Caesars Forum Shops  and then back to the Palms. The Palms shuttle is available  to all.

The Hard Rock Hotel :
The Hard Rock Hotel and Casino provides free  shuttle service to and from the Strip on the hour from  10:00am to 5:00pm.  It departs the HRH valet on the hour and drops off at the Fashion Show Mall  valet by Forever21.  It departs Fashion Show Mall at 15 minutes after the hour to return to the Hard Rock. It is intended for hotel guests, but anyone can use it as long as there is space available.

Coast Casinos  (updated April 26, 2016):
The Coast Casinos run a shuttle between  the Gold Coast (South entrance), the Orleans (South entrance) and the Strip (High Roller  valet). They run every 30 minutes from 9am – 12:30am. The shuttles are free to everyone, but Gold Coast and Orleans hotel guests receive priority. The last shuttle leaves the Forum Shops at 12:30am.
The Gold Coast is across the street from the Palms, so this shuttle is a good alternative to the Palms’ shuttle.

Palace Station:
Palace Station runs a shuttle from Palace Station (Main Entrance) to the Fashion Show Mall (Main Entrance). The shuttle departs Palace Station at 10am, 11:30am, 1:30pm, 3pm, 4:30pm, 6pm, 8pm, 9:30pm and 11pm. It departs Fashion Show mall 15 minutes after departing Palace Station. You can download a PDF of the schedule from their website (scroll down past the room info).  The shuttle is officially provided only for Palace Station guests (it is part of what the resort fee covers), but Boarding Pass players club members have been allowed in the past.

M Resort:
The M Resort runs a shuttle to the Fashion Show Mall 4 times a day at 10am, 2pm, 6pm and 10pm. The shuttle departs Fashion Show Mall and returns to at 10:30am, 2:30pm, 6:30pm and 10:30pm. The shuttle is free. M Resort guests receive priority.

Red Rock Resort:
Red Rock Resort runs a shuttle from its valet entrance to Fashion Show Mall (North entrance, next to Capital Grill) every 2 hours from 11:10am to 9:10pm. It returns every 2 hours from 11:40am to 9:40pm. The shuttle is for Red Rock Resort guests, but usually allows non-guests when space is available.

Sam’s Town :
Sam’s Town runs 1 shuttle to the Strip and another to Fremont Street/downtown.

• The Strip shuttle departs the Harrah’s valet at 9:30am, 10:50am, 12:10pm, 1:30pm, 3:40pm, 5:00pm, 6:30pm, 8:00pm and 9:10pm and takes about 30 minutes to get to Sam’s Town. The shuttle departs Sam’s Town and heads to Harrah’s at 10:10am, 11:30am, 12:50pm, 2:10pm, 4:20pm, 5:50pm, 7:20pm and 8:40pm.
• The shuttle to downtown stops at the Fremont Hotel (Valet entrance) and California Hotel (Lobby entrance). 
This shuttle runs at a more predictable 1 hour and 15 minute interval. It starts at Fremont, arrives at California 15 minutes later and arrives at Sam’s Town 25 minutes after that. It departs Fremont at 8:15am, 9:30am, 10:45am, 12:00pm, 1:15pm, 2:30pm, 3:45pm, 5:00pm, 6:15pm, 7:30pm and 8:45pm. The shuttle departs Sam’s town at 9:00am, 10:15am, 11:30am, 12:45pm, 2:00pm, 3:15pm, 4:30pm, 5:45pm, 7:00pm, 8:15pm. Sam’s Town and Boyd Gaming hotel guest who show their room key will receive priority.

Silverton:
The Silverton runs a shuttle to the Forum Shops at Caesars Palace departing at 10am, 1pm, 5pm and 10pm, and depart from the Forum Shops back to the Silverton 30 minutes later. The shuttles are reserved for Silverton hotel guests.

There are also 3 free Trams that run from TI to the Mirage, Bellagio to the Monte Carlo with a stop at Crystals/Aria, and Excalibur to Mandalay by with a stop at the Luxor.

Disclaimer: Hotels occasionally adjust their shuttle service times and have even changed their pick up locations on occasion. I verify and update this information regularly, but it is worth contacting the casino directly if you are going to depend upon the shuttles.