Equipment is largely irrelevant in terms of rushing - while you're rushing, if you can't handle a monster, just back off and let others kill it. In general, unless you have someone specifically rushing for you (as mentioned in these other answers), you'll only have to worry about waypoints and the staff in Act II. You might need help with the Ancients and bosses, but you can usually find a game for that. So if you can get someone to give you the waypoints you need (and the staff...), all you really need to worry about is where you're receiving the best experience. For the best experience with very little resources, I follow this procedure:
- Levels 1-15: Normal Tristram runs
- Levels 16-20: Normal Tomb of Tal Rasha runs
- Levels 21-24: Normal Cow Level or Chaos Sanctuary runs
- Level 25: Normal Ancients to gain 1 level, achieving level 25
- Levels 26-50: Normal Baal runs (somewhere in the 40s people usually transition to Nightmare - you can't do Nightmare Ancients until 40, but you get decent XP from Normal Baal runs until about 50)
- Levels 51-60: Either get through Nightmare and do Ancients then Baal runs until 60, or get through Nightmare and Hell and do Hell Chaos runs
- Levels 60+: Hell Ancients, then Hell Baal runs
Alternatively, some people get bug-rushed all the way to Hell and then do Hell Chaos until level 60.
As far as rushing goes, you can often get someone to rush you through a difficulty by offering them the drops from your Forge (Act IV, Quest II - destroying Mephisto's Soulstone).
There used to be a concept called "ubering" (which is the most expensive, but also the easiest). It involves getting a few high-level characters together to kill all the Ubers in Uber Tristram except Mephisto, then get him stuck in a house while minions continue to spawn. If you get a few Hammerdins and their mercs together, you can basically sit AFK while they kill everything (by holding down Shift + Left Click, using a script/bot, etc.). This allowed for 5-6 low level characters to gain significant numbers of levels quickly. The explicit process is available here.
Ubering (also called uber leveling or UT leveling) is no longer available as of 1.13 (it appears Mephisto's minions no longer grant experience). There are rumors of it being reinstated in 1.14, but it is unlikely.