Research is an exciting journey, the results are fantastic (most of the time anyway).
First dig a hole, find some crystal shards, make clusters out of them and decorate the area around your bookcase with them. Oh, you didn't have a bookcase? Every self respecting mage has his research table accompanied by a bookcase!
Then you'll need a way to get those pesky research points. There are multiple ways to getting them. You can grab your thaumometer and scan everything, and every living being, don't forget to scan the dead too!
Tired of risking your life having those creepers explode before you get a chance to scan them? Don't worry! You can always try looking for nodes.
But what do you do when you can't find nodes? You ran out of items to scan and all your research points are gone? Your boots are gone from falling down all those cliffs and trees? You have more arrows then hair on your head? Don't worry!
Remember these clusters I told you to place near your research table? When you get back from your nodehunting, your reasearch table might look like this:
You might ask what that black stuff is, that's my fault, I dropped a fresh inksack all over it. But fear not, that ink is not important! The important parts are still visible!
You can see how Ignis and Aqua are all blinky? And Aer is not? Now this is why I told you to place these bookcase and clusters all over the place, not for decoration. They generate research points! Granted,
only primal aspects after more research it seems that any research aspect is eligible for free bookcase research points, only one of each aspect per research table, and awfully slow. But it works, kind of. Aqua and Ignis have a hidden research point, Aer doesn't.
There are multiple specifics to them.
First thing: they are bound to your research table, and not to the researcher, so breaking and moving your research table will destroy them.
Second: they are always the last thing you'll use up.
So in the table shown above, I'll first need to use up all 107 Aqua first. Then there'll be one more Aqua left, that's the table bound one. Use this for research, or combine it to create some other element you need. It's really slow going, but it'll help if you ever succeed in running out of research points.
Third and final point: they are generated by crystal clusters and bookcases in the area around your research table. The clusters will generate their respective element, so an Ignis cluster will generate Ignis, an Ordo cluster will generate Ordo and so on. A bookcase will generate a random aspect point.
Still running out? You can discover something that might help you get more research points. I've hidden it here to not spoil the surprise of finding it yourself.
At a certain point you can research into the magical workings of a "Deconstruction Table", this will allow you to break items into their primal compounds. You only get a maximum of one primal compound per item, there is no guarantee that you'll get anything at all, but you can put as much in as you want. Items with a higher complexity will result in a higher chance of getting a primal research point. You can put a whole stack of items into the table at once to be broken into parts one by one, but the process'll stop once you find a primal aspect. And restart again when you click the aspect to add it to your available research points.
Thanks RavenM for having a larger answer about the first two (most efficient) ways of getting points, so I could mostly skip over them in my answer.