This assumes you have already picked an agent, or more likely that someone else is getting the missions. If you do need to find yourself an agent you can find one using the guide here. If you are getting the missions you also need to work on your social skills: look them up in the Market to see which ones have an impact on your payouts and loyalty points (LP), and your standing with the agent. Also bear in mind that if you are missioning for LP the LP store of your chosen agent's corporation better have something you want.
Once you have a mission, you need to pick a ship. Eve ships come in all shapes and sizes and you might haver a number that you have set up for specific tasks. The PVE ship needs to be specialised in two things, and two things only; Tank (being able to withstand a lot of damage) and Gank (being able to cause a lot of damage)., in that order. It's no good using a cloaking ship, or an ewar ship, or a ship with a huge hold, or a ship with a massive range. PVE combat missions are, for the time being, very simple. Get in, withstand the damage long enough to kill everything, get out again. Anything else on your ship is a waste.
All the races have ships that are suitable for PVE combat but some are easier than others. It is interesting to compare the damage output from NPC ships as a relative guide to what works best, and on that basis the Typhoon wins out, but the Typhoon is hardly ever used for PVE. Why? Because it takes a lot of real life skill for the micromanagement of shields, capacitor and weapons systems to make it so, and a lot of ingame skills to make the uber fittings work.
The ships that are used most are used because the more ingame skills you acquire, the easier they become to fly. Don't be put off using your chosen race's ships because you have been told they are 'worse' than another though. These things are purely relative. However check the ship info to see what bonuses it gets. If you are looking at a ship with bonuses to both Tank and Gank and nothing else, that's your baby.
I don't want to go too much into ship setups here, as that is a massive subject that people argue about on the forums every day, but a note on resistances: One of the main differences between PVE and PVP is the predictability of the enemy. You can look up guides to missions here, and with practise you will know them backwards anyway. This means you can prepare hardeners in advance to boost the resistance of your shields to the specific damage type and level the enemy will be throwing at you.
You can use a shield or an armor tank for PVE but shields are usually considered easier to deal with, as they use up those mid slots and whichever you use make sure you leave some low slots open for damage increasing mods. You can also utilise a very high shield recharge rate and resistance on some ships, Drake for instance, to put together a 'passive' setup that means you don't have to touch your defense mods at all, and can concentrate on hitting things. Again, for more on passive setups try the forums. To Repeat: This does not mean that you cannot use an armor tank for PVE. A friend uses a Megathron, for instance, very effectively.
If you are just helping out on someone else's mission, then assuming the person leading the mission knows what they are doing they should be the tank so you should be able to set up purely for gank; however for the sake of careless accidents I would always fit some tank, even if you never use it. In this circumstance you only need enough tank to keep you alive the couple of minute or so that it takes you to warp out. A damage control 2 makes a very useful emergency tank. The leadership skills of the fleet commander may also help. Bear in mind that speed mods, like afterburners, also count as Tank (ie running away), and are useful just for speeding up the mission too.
Once you have your ship set up, then, it's time to set out on your mission. You will usually recieve an invitation to join a fleet, either as an orange link in channel or a pop-up box. A couple of things to be wary of here, from the not so friendly; firstly your commander in the fleet has the ability to warp you, if he is in the same system as you, and could warp you into a trap. Secondly it is just about possible, by manipulation of the war system, to make you susceptible to being blown up even in hisec. Make sure you trust the people in the fleet you are joining.
You will get two new windows pop up when you accept; the fleet window and the broadcast window. Once you are in the fleet you can safely minimise the fleet window. Put the broadcast window somewhere out of harms way for now. I usually have my overview top right, my stack of chat channels bottom left, just above the row of minimised windows, and where it doesn't encroach on the HUD buttons. You might want to close a few windows you aren't using, it will speed up your connection and be less distracting. I put the broadcast window, made as small as it will go, just on top of the chat window. I then pin all of the windows to make them semi transparent, and keep them from floating around too much. Finally to both reduce lag and make seeing what is going on easier I turn on the tactical overlay, click both mouse buttons together and pull back to see the wider arena. For support vessels, rather than the tank, dont pull back too far, it is useful to see who is shooting.
So the tank warps in first, makes sure thay have at least some of the aggro (that is, the damage coming from the enemy) and calls you in. You will have a number of NPC (non player-controlled) ships, signified by red crosses of different sizes. In the overview it will tell you what these crosses mean. Sort your overview by distance from you and right click on the first one, and target it (or ctrl-click). right click it and click 'look at'. If it is firing at something you are safe to fire at it without fear of reprisal, NPC ships never switch targets unless their original target dissapears. DON'T use drones unless you are told it is safe to do so, and if you do then right click on the top left of the drones window and set them to passive, so they won't go off causing aggro somewhere unexpected; if a drone gets aggro, and then dies, the NPC will look around for something else to shoot at, and it could be you. Stick with the fleet commander (orbit them closely or just right click on them and click approach at the start) unless they tell you it is safe to do otherwise, proximity to the NPC's may likewise trigger aggro.
A knowledgable fleet commander may pick targets for you to shoot at; these will be marked by commands to 'Target!' and cross hairs on the ship in question in the main window. Either can be right clicked for targetting purposes.
In most missions specific ships are the trigger for new spawns of enemies, and the fleet commander will tell you to avoid shooting them. Those ships aside, you should either start with small ships (frigates and cruisers) and work up, or if you are being assigned targets by the fleet commander you may collectivevly be shooting at the largest ships and working down.
Missions are usually done on a casual basis so most of this will be communicated, usually, in the fleet window, rather than using audio communication. This is both so it stays there, for reference, and also because not everyone in the fleet may have audio access.
Follow the text commands otherwise, heading through warp gates AFTER the tank until the mission is complete. If you are in a position to loot or salvage a mission (ie you have a suitable ship and the mission owner has said it was ok) bookmark each gate as you come to it.
After the mission has been turned in as completed to the agent these gates will dissapear, but the loot and salvage will not, so you can return to the bookmarks unmolested to pick through. You may wish to rename them as numbers as you save them if you have been doing a lot of missions; you have up to two hours to return to them and they dissapear in order (that is, the bookmarks will eventually drop you in empty space). Salvage and loot after a mission can effectively double the total value of a mission, so do not dismiss doing this unless you are very rich or very short of time.
Your share in the bounties (an even split, if you are in the same area where they were killed) will arrive approximately twenty minutes later, and your share in the mission reward will come immediately it is turned in if the fleet commander decides to do so. At this point run away cackling, and counting your isk.