Other nice healing addons are Grid (gives a quick overview of large groups), Healbot Continued (like Clique but more complex), and Paladin Healing Calculator (helps you figure out what gear will be most beneficial.)
Another important role for pallies is blessing your allies, and there are a few addons that will help with that. I use Pallypower, which enables me to quickly see and change the greater blessings of my party members (and even their pets!) It decreases a lot of the "5 minutes left on Might!" chatter when they know that you have an exact timer and warning for when they're going to need a re-blessing.
I'm one of those people who think that every paladin should have gear for tanking and healing, as well as the more popular DPSing. You should be able to grab a decent set of healing gear just by picking healer quest rewards when there are no good rewards for your spec. Since plate healing gear is somewhat lacking pre-BC, it's perfectly acceptable to collect some cloth gear pre-Outlands. Anything with plus healing or plus damage/healing, even if it's not plate, is something that you should pick up and store for when you're called on to heal -- but you'll most likely be finding intellect, spirit and stamina as your main stats.
Once you get to Outlands, you'll find a wide variety of great healing gear. Hellfire Peninsula's early quests have some rewards that should last you quite a while as a backup set. Once you start getting the good quest rewards, you can be a little more discriminating in your stat choice. Look specifically for healing, spell crit, intellect, and mp/5. Store them in your bank in a way that they're easy to find when you need them.
Technically, you can heal in your soloing gear, but I wouldn't recommend it. If you're a prot grinder with a lot of dam/heal gear, you're going to do tons better than a retadin with mostly DPS warrior stuff -- and your group will notice if you're still in +strength gear.
How to do it
All things considered, paladin healing is not very difficult. You've got two main options -- Holy Light, a 2.5 second cast that can heal for a huge amount of your target's health, and Flash of Light, a 1.5 second quickie that heals from a fourth to a fifth of the amount healed by Holy Light. However, Flash of Light is much more mana-efficient and is the heal of choice for most paladins. (Holy Light is good for healing after a fight, or if your tank is taking heavy hits and needs to get up by a lot of HP.) Lay on Hands is an interesting little spell that heals for your full mana pool, costing, of course, your full mana pool. Do not bind it near anything important, and save it for when the boss is almost dead and the tank's about to bite it.
And ... that's about it for non-holy pallies. Holydins will have Holy Shock, an instant-cast heal that heals for about the same amount as Flash of Light. They also have talents to increase their mana conservation, crit chance, etc. This scarcity of spells means that paladins are best as single-target healers. If you're lucky enough to find yourself in a group with a second healer or hybrid, ask if you can take the tank while they spot-heal the DPS.
But being a healer goes beyond just healing. Paladins can Cleanse off poison, disease and magic effects, making certain instances a lot easier. Keep an eye on the debuffs of your party so you can take care of those pronto. Blessing of Protection can transfer your "bubble" to a DPSer that's pulled too much aggro. Unfortunately, the mobs sometimes decide that means you're the next person on their hit list. Use it carefully and make sure there's a tank-type person between you and the DPSer. Blessings have been covered extensively elsewhere in this column, so all I'll say today is that you should bless the tank with Light if at all possible.
Other than that, just act wisely. Stay as far back as possible if aggro is a concern. If you have a very good group, ask if you can run in and judge the mob occasionally if no one needs heals. And watch out for ninjas!