I havent seen any quick start guides to all of the build choices, skill point modifiers, or items yet so ill try to make a simple guide that should work for about the first 15 levels or so and get you leveling with the least downtimet. I wont be going into great detail, this is only meant to introduce you to STO quick enough that you can get into some combat, not die, and get your fair share of loot in fleet encounters.
Skill points work on several aspects of gameplay. They will affect the overall qualities of the ship you fly and buff up your abilities making them more effective. That could be increasing the duration of an effect, reducing the energy consumption, or increasing the output of the ability/weapon. Some of the first tier skills will instantly change how much damage you can do with your weapons while others simply buff a bridge officer skill. I find the later to be of a lessor concern when starting out.
When first leveling you want lots of DPS. The starter ship has room for playing around with configurations and you dont have to be as worried about ship configuration, or weapon choice because the reactor will support all 3 slots just fine. So early on dump all your points into these skills (i have also found that this order works well)
Starship Energy Weapons - to 9
Starship Projectile Weapons- to 9
Starship Command- to 9
Starship Warp Core- to 9
These skills have a direct impact on how much Damage you do, how well your ship can turn, and how much power you can get out of your ship.
A word on Bridge officer choice. Make sure when you start out that you have a bridge officer with emergency power to shields. IMO this is the most critical skill to have right away. It will enable you to recharge your shields quickly after a good fight, or survive an alpha strike (when you agro all 8 enemy ships and they focus fire on you). Speaking of alpha strikes, they happen, so dont get too worried. You also dont have to die every time. Quickly divert all your power to shields, hit emergency power to shields and start sending power to the shield that is taking the most damage. Start turning in circles to distribute the damage across all your shields and wait for help to arrive. In some cases you can soak up the damage and slowly pick off the ships starting with the smaller ones first and working to the battleships. It helps if you go in and modify the default power settings for defence. Crank shields and aux power up, crank weapons down, and give engines a little love but not much.
I have my opinions on how to level quickly and what works well for me. You need to decide what your favorite playstyle will be and what skills you want to use when making further skillpoint choices. I will say from what time I have spent in the game; DPS is king. SO after you buff up the above skills to level 9, spend some points where they will improve your bridge officer skills (emergency power to shields, tach beam etc.) Then head back into the DPS trees again when you get your Lt. Commander rank. Just stick to a single weapon type, or at most pick 1 energy weapon and 1 projectile weapon to use on your ship. THis way you get the most bang for your buck out of the skills, and the items in the game.
Outside of skill points you need to also be thinking about what consoles you are going to run. Always look for ones that increase power output or give a plus to your weapon damage. Think about your choices carefuly because running lots of beam weapons means your hitting your reactor hard all the time. When your power level drops your dps goes down. This becomes more of an issue in your second tier ship than on the starter ship.
I will also recomend that you do plenty of explore type missions, and enemy signal missions. These seem to give you what you need for the best gear and level you quickly. You are going to need badges to buy some higher level gear, and you will get good drops from the battleships and battle cruisers in the deep space encounters.So if you stick to these play areas you will have the most options to upgrade your ship (if you dont want to wait on drops or hope you get them from missions) I dont find that missions are worth the time investment right now. If you want some story line and a clue as to what is going on in the universe you may want to do missions.
Sell everything that is not of MKII quality or better to your ships replicator. When you have several thousand credits go to the market and buy the MKII version of any ship fitting that you do not allready have. DO the same when you get your Lt. Cmdr rank. Sell all your MKII stuff and be on the lookout for MKIII and MKIV gear. IF you have enough money buy all highest rank gear you can afford and put it on your new ship before you leave the dock.
As a rule of thumb - sell your consoles to private partys for the most credits. Good ones are hard to find and fetch a great price. The ones that add to weapon damage sell for the most. Fit one that matches your weapon choice and sell the rest for mega bucks. Shields, and weapons also fetch good prices. Sell all the rest of the space crap you pick up to your on board replicator. When you have an inventory full of consoles, weapons and shields go to the market and sell them.
I havent found a need to spend any skill points on ground based abilities yet. Not within the first 20 levels anyway. It is entirely possibly to never do a ground mission if you dont want to, and level up.