Everquest has a decent map, so use it. Backspace or "m" should bring up the map window; notice that it has two tabs, one of which is an atlas, use it to locate where zones are. Samanna also keeps a very useful zone interconnection map here, although it hasn't been kept fully up to date.
Make sure that you have installed the maps from mapfiend, otherwise the individual zones, apart from newbie zones (and not all of them) will appear empty.
Wizards and druids have a set of port spells that can take you close to most zones; if you play one of these, or have a friend who does, you should be in good shape for getting around.
There are also six different systems of ports that anyone can use, of which three are in common use. The first is the set of stones in Plane of Knowledge, each paired with a book in the various newbie zones. If you are familiar with the general map of Everquest, this is very effective for travel.
The most useful set is probably the Guild Hall portal. You buy an item from the gnome vendor next to the portal, and give it back to him. That sets the portal to your destination for any traveller until the next person decides to reset the portal (or the zone has to reload). This has more destinations than other systems, and new ones are added from time to time. If you want to use this, you have to join a guild - which you should do anyway.
The next is the set of maguses introduced with Lost Dungeons of Norrath. There are six of them in newbie zones in Antonica, plus one in the Guild Lobby, and ones in Abysmal Sea, Nedaria's Landing, and Natimbi. The GL magus is one way only, and the three in GoD zones are separate from the Antonica ones. You must have done the quest to get an Adventurer's Stone to use the main Antonica magus ports; you can reach Nedaria's Landing, and thus Abysmal Sea and Natimbi, without one.
Next are the Luclin spires, which are hardly ever used now, and just mentioned for completeness. There are spires in the Nexus (on Luclin), Greater Faydark, North Karana, Toxxulia Forest, the Great Divide (Velious) and Dreadlands (Kunark). They only port every 15 minutes, which can seem kind of slow. To use the Nexus port, just stand on the pad for your destination; to use any other spires, talk to the NPC to get a shard.
Not many people know about the next one: in Timorous Deep, in the SW corner underwater, there is a room with portals (firepots) you can click to port to the main cities in Antonica, Faydwer, Odus and Kunark. If you can port out (ie, are a druid, wizard, or have the Origin AA), you can bind there usefully. It is just barely conceivable that the rare spawn dragon in Timorous Deep (Faydedar) can find his/her way into this room, so it's not a great place to go AFK for a long period.
The last portal involves the Priests of Discord. Any of the Priests of Discord can port you to Dranik's Scar, and one of the priests where you arrive will port you back to where you came from. There is a little glitch involving Crescent Reach which can be used to port new characters from Crescent Reach to the Plane of Knowledge.
There are several other portal systems that I haven't mentioned: one in the Hole, to Neriak and Paineel (and somewhere else); and others in Veehsan's Peak and the Plane of Time.
And finally, boats. There are boats between various locations. Because they have historically been buggy, some of them have been replaced by transporter gnomes; however, some zones are still best/only reached by boat routes. Examples are: Erud's Crossing, Timorous Deep, Ocean of Tears.