Things To Do In Eve With (almost) No Skilltraining

Do the Rookie Missions: They pay pretty well, mainly because they give you some starter ships and skills, there are 5 sets of rookie missions for each race and 3 sets of agents per race giving you 12 tutorial systems in total. You can run all of them with only a couple of extra skills required. While you're at it, the Sisters of Eve arc is pretty profitable for a starting pilot, 10million isk from all the rewards plus the loot. After all those missions are done you can consider level 1 missions from your favourite faction, but don't expect the rewads to be nearly so good.

Mine: don't bother, unless you're going to be AFK for a while, there's better ways to make money. A noobship with a full hold of veldspar will make you 10,000 isk if you're lucky.

Kill NPC Pirates (Rats) in Asteroid Belts: Strap some guns on your ship and go to an 0.5 security status system, the rats are easy and get you 7000-13000 isk each. The loot the drop is mostly crap but can be awesome, pay attention to anything dropping a tag, tags can be worth millions, especially copper tags that tend to get ignored by mission farmers because they're in frigates. Also pay attention for commander spawns - Dread Guristas, True Sanshas, Domination, Shadow Serpentis and Dark Blood - these can sometimes drop epic loot like pirate implants worth 100million isk.

Run 1/1 Plexes: You'll see these on your overview, "Sanshas Military Outpost", "Old Meanie Cultivation Center" and similar, they're always in the same places and contain a whole bunch of weak targets that respawn once an hour. In the final room you'll find something that drops awesome 'deadspace' loot, it might be a can, a ship, or a structure, and you'll get things like 'Centii C-Type Small Armor Repairer' which can sell for 10million on contracts. You'll most likely find that someone has beaten you to it maybe 75% of the time, so just keep moving and go from one place to another picking up the fancy modules. It's pretty easy to pull down 100million in modules in a couple of hours using a pilot with only 12 hours of training to fly a combat frigate.

Do all of the above in low-sec/0.0: Because you have no skillpoints and a basic ship, you have very little to lose, just be sure you learn the magic 'spam warp button' trick to escape if you get tackled and killed. And, insure your ships of course. I took a noob character into Delve when the Goons were kicking out BOB and lost about 60 noob ships, which cost me nothing in the long run, except for crappy killboard stats.

PVP With 'Friends': Train the 'propulsion jamming' skill and you can be an asset to a PVP gang, solo pvp isn't going to happen unless you're ridiculously lucky and find someone in a defenceless ship away from sentry guns. Stick a warp disruptor on a rookie ship and you can stop victims from escaping while your friends kill them. With a little more training you can get in a combat frigate with a microwarp drive and kill things as part of a team, there's a great video showing 2 pilots with less than 150,000 skillpoints flying around 0.0 and killing the locals with great success. Or you can just join Red vs Blue and die lots while getting on the occasional killmail.

Trade: Setup in your local trade hub and watch the markets, buy low, sell high, maybe move the odd pile of gear to or from the hub. The more money you have in the market the faster you'll make money from the market, you can train a few trade skills for more order slots but no training is required to get started.

Take advantage of other people's Hard Work: Profit from other people's wrecks, you'll find them all over belts, especially around Hulks which are AFK with drones out killing the local rats. They'll get kill rights on you, but only for 15 minutes, and it's highly unlikely anyone in hi-sec will have warp disruptors, so just warp out as soon as you get shot at. Do it in a rookieship and then getting killed won't cost you anything.
Many of the best wrecks in hi-sec are in missions, so train astrometrics 1 and you can scan down battleships running missions.If you're really lucky it'll be some multi room mission and the pilot will be dealing with enemies deeper in the mission and therefore no threat.
Train up the salvaging skill (about 4 hours for a noob to get the pre-requisites) and you can instead just salvage the wrecks and the pilot won't get kill rights, he might get mad but he can't do anything unless you steal stuff from inside wrecks.

Take Advantage of other pilots Misfortune: There's always wars going on and if you sit outside a trade hub, or even at some hi-sec gates you'll see other pilots getting losing their ships, again, steal from this and make a mad dash to station with your loot. If you have the salvage skill pay attention to any 'elite' wrecks, these yield tech 2 salvage components which can be worth a lot.

Take Advantage of Dumb Players: Scam contracts (doesn't work on trial), I could write a book on scams, suffice to say if you're in Jita and you see a contract posted it's probably a scam, and you should open it up and try to figure out what the scam is just so you don't fall into it, and can perhaps exploit this later. I've seen people successfully sell a piece of Carbon for the price of a Charon freighter.

Carve Your Own Path: because there's always new things to discover.