How it Works
In Career Mode your objective is to fill out the Tech Tree by gaining science. These are some strategies and explanations on how to be successful.
Science points are gained by doing experiments. The amount of science gained for each experiments depends on the location, type of experiment, and times you have already performed the experiment. The type of experiment is determined by the equipment you use. The location depends on the biome, doing an experiment on the same biome will not yield different results. The more times you have used the same equipment, on the same biome, the less value there is in each experiment (this does not mean doing the same experiment more than once is not useful.)
Science points also scale depending on distance away from Kerbin. Experiments performed in orbit around Kerbin generate more Science Points than those performed at Sea Level, escaping from Orbit around Kerbin generates more Science Points than those performed in orbit, and so on.
How to Experiment
To experiment you must right click on the desired equipment and left click on: Crew Report, Run Atmosphere Analysis, Observe Materials Bay, Observe Mystery Goo, Log Temperature, Log Pressure Data, Log Gravity Data, Log Seismic Data, EVA Report, or Take Surface Sample. Once you have experimented you may transmit, or keep the data. Transmitting will make more space for further experiments, but requires a communication device and Electric charge. If you keep the data you may transmit it later, or keep it and recover it with the vessel. Note: Transmitting data will not allow the equipment to make further experiments, except for EVA Reports and Surface Samples (however you may not store more than 1 of each experiment in the module).
Bear in mind that the Materials Bay and Mystery Goo are single use experiments. You may transmit the results at a greatly reduced yield. You can also return the results to Kerbin for full science credit either by returning the experiment itself or moving the results to the command pod by EVA. A science lab may be used to reset single-use experiments and will also allow transmission of results at a much higher yield than is otherwise achievable.
There are many ways to play Kerbal Space Program, but these are some general guidelines for Career Mode.
Walking around the Kerbal Space Center and taking Surface Samples of the Launch Pad, Runway and the grass around the Space Center all generate Science Points.
Launch "rockets" consisting of only a command pod. You can use it to take a crew report from the launchpad. Then you can get the pilot on an EVA and take EVA reports and surface samples.
First Sub-Orbital flight
On your first launch you may want to get enough science to get the next tech tree node Basic Rocketry (in particular the TR-18A Stack Decoupler) to build more efficient rockets. A simple rocket with a single FL-T200 Fuel Tank, LV-T30 Liquid Fuel Engine, Mk16 Parachute, and Communotron 16 is enough. While on the Launch pad perform a Crew Report and transmit. Then (without falling off) perform an EVA Report, and keep it. Then after liftoff, perform another Crew Report and keep. Then once you touch down (you will certainly break the engine), take a surface sample from the biome you landed in and recover the vessel.
First time in space
Now that you have the stack decoupler, you are able to build a multi-staged rocket powerful enough to lift you out of the atmosphere.
Take the rocket from the sub-orbital flight. This will be the third stage of your rocket. Add a TR-18A Stack Decoupler below its engine, followed by the newly discovered FL-T400 Fuel Tank and another engine. This is the second stage. Then add another decoupler and a RT-10 Solid Fuel Booster as the first stage.
After liftoff, wait until the solid fuel booster has burned out. Then press space to drop it and space again to activate the second stage. When the first fuel tank is empty, press space again two times to discard the second stage and activate the final stage. This should bring you on a trajectory high enough to get a "Crew Report from Kerbin's upper atmosphere" (transmit) and a Crew- and EVA-Report "while in space near Kerbin" (keep). When you feel bold you can also attempt to take an EVA report from Kerbins upper atmosphere, but make it quick or your Kerbonaut will get blown away when the atmosphere gets too thick.
After you landed (likely in the ocean east of the space center) you can also take another surface (water) sample (yes, your Kerbonauts can swim).
For the proper audio background during the flight, listen to this youtube video.
Your next goal is to explore Kerbin. Kerbin has several biomes. Each one can give you a surface sample, a crew report from flying over it (usually obtained during launch and re-entry) and an EVA report from orbit above it. One method to obtain reports from many biomes is a small spaceplane that flies between biomes carrying a Kerbonaut, several Mystery Goo canisters (optional), and enough Electric Charge to transmit all the Crew Reports earned from the flyovers and landings. You may (this is recommended) add as many other science instruments as you like with the possible exception of the SC-9001 Science Jr. Materials Bay. The required parts to do this can be earned by researching the Aerodynamics node. (I have not tested this myself I'll tell you when I have.)
The last technology also unlocked the Mystery Goo Containment Unit which is an experiment you can perform under different conditions and which also gives you a considerable amount of science points.
Your next missions should be to obtain these science-points.
There are two limiting factors for the science points you can obtain per launch in this phase: the limited amount of electrical energy in your command pod which limits the number of transmissions you can perform per launch and the limitation that some experiments can only used once in a mission. You can circumvent the first issue partially by launching rockets with multiple command pods, but this doesn't scale well due to the high mass of them. So you should focus your research on Survivability so you can research Science Tech, which unlocks the Z-100 Rechargable Battery Pack. The next important tech node is Electrics, because it unlocks the OX-STAT Photovoltaic Panels. When you put these on your rockets they can restore their electricity reserves during the flight, which practically allows you to transmit as often as you want and makes recovering of scientific reports unnecessary.
For single-use experiments, you can attach multiple of them to your spacecraft. Again, this may not scale well for some due to weight (i.e. the materials bay), but can work well with others (mystery goo). For single-use experiments, it's best to return the results to Kerbin. The results can be removed by EVA and placed in the command pod. Use only one copy of each experiment in each experiment location as multiple copies of the same result cannot be stored in the command pod. Typically, one returned experiment will exhaust most of the science points available.
Kerbin is good for getting some bits of science for unlocking the lower-tier nodes, but you will soon find the need for larger portions of science. And these are waiting for you on the Mun. Did you know that the Mun has even more biomes than Kerbin? And did you know that each one of them has a higher science multiplier than anything you can do in Kerbin orbit?
When you do a Mun mission, you should thus try to visit as many biomes as possible per mission. Building a rover might seem like the obvious solution, but navigating a rover on the Mun is harder than it seems and getting from biome to biome takes much longer than you would expect. A much less time-consuming way is to construct a lander with enough fuel that it is capable of performing multiple sub-orbital "hops" between different biomes before returning to orbit. However, the limiting factor will be the number of SC-9001 Science Jr and Mystery Goo Containment Unit you can carry. You can solve this problem by bringing a Science Lab down with you, which will allow you to reuse your experiments, but the high mass of the science lab will then cause fuel to be a limiting factor. Another method is to keep a science lab in orbit and rendezvous and dock with it between hops; this requires good proficiency with orbital mechanics and docking.