On a kyu level, especially up to 10k, the most important thing to study is tactics. Strategy is much more difficult, and can be reserved for study at dan level.
There are several ways to study, but with the Internet, one possible way to improve is just to play, and to play a lot.
About the methods and exercises below, most important advice is: Do one exercise at a time. In go, learning one thing at a time is likely to make it more fun, and progress faster. Trying to learn everything at the same time will just make it more frustrating, and harder to see what you do wrong. And, experiment. Try new things! It is the only way you can learn.
Undoubtedly the most important study method, especially if you have players around, or slightly above your own strength to play with. Playing both slow and fast games works well, and which is better for you is probably mostly a matter of which you like better. However, the following advice is usually given to new players: "If you play faster games, you have time to play more games." Insei and Japanese players in general have a tendency to play much faster than the games you would see on a regular club night at a European club.
Keys: Take the games seriously, and try to learn something from every game, even if you are playing quickly.
Improves: reading, intuition, counting
Getting teaching games, either at a local club, or on the internet, is a good way to improve.
Particularly, it enables the teacher to give targeted advice to the student, that you can't get through any other study method. The same is true for reviewing your own games with a stronger player.
Keys: Take the games seriously, and take special note when the teacher makes a move you don't understand.
Improves: all aspects, understanding of key concepts/ideas
Problems, particularly life and death problems, is important, not only for the sake of learning to solve the problems, but also because it makes you stronger at reading, shape sense and tesuji. Most strong players recommend solving simple problems. Those are problems you can usually solve within 10-20 seconds, and if you don't get them in 1 minute, you flip the page to look at the solution. There is an important reason for this. At a basic level, the reason you don't get a problem is that you have misunderstood some fundamental idea/technique. Looking at the solution is the fastest, if not the most fun way to remedy this.
Keys: Try to solve each problem completely, without any doubts. If you can't do this, you are either not trying hard enough, or the problems are too hard.
Choose a problem collection that is easy for you (Graded Go Problems for Beginners volume 2 as an example) and go through it. You may take 2-3 sessions for it, but try to do it in a concentrated form. Record how much time you took for all the problems together, and your success ratio. (Be honest, only give yourself credit to the problems you solved without a doubt, don't be content with finding the first move, say!) Repeat again a few weeks later to test your improvement. Be careful to make sure you are solving the problems again, not remembering the solutions.
Improves: shape sense, reading
Other types of problems are worth studying too, most notably, tesuji and opening problems. Endgame problems can be useful too at a somewhat higher level.
Same as for 2a
Improves: shape sense, reading
This is important when you start, especially if you don't have any teacher, but as you progress, it becomes less and less important. Simply speaking, books can tell you what a private tutor would, but not targeted specifically at you. However, there are around 10 books in English that are well worth reading. Particularly, books are good to read when you would not normally study go otherwise. On the train, before going to sleep at night. The same is true for problems.
Vincent: Do you have some specific recommendations when it comes to books?
1. Try to apply what you read in a book to your games. Be it tesuji, attack and defense, or endgame, try to spot and apply similar ideas. If you manage, great! If you don't see any situations like that, it is likely the material is not very important for your progress. If you feel you see similar situations, but can't apply the material, you are probably on the right track, if you keep trying, you will probably succeed.
Improves: all aspects
For strong players, this is probably one of the best ways to improve, but it is useful also for beginning/intermediate players. It is important to study games on a real board, away from the computer. This will greatly enhance your "experience" of the game, and increase your concentration.
Try to memorize a pro game. The first times, 25 or 50 moves is enough. Then, increase to 100 or 150 moves for each game. Usually, you may stop looking at the game as the endgame begins, around move 150. In the beginning, try to choose simple games, without long ko fights etc. Later, you can choose more difficult games. To memorize the game, play it through once. Try to have a feeling of the moves, rather than to try to understand the reason for each move. Then play it through again, until you think you can memorize the first 25, 50, 100, etc. moves. It is enough to play through as many moves as you are planning to memorize each time.
Improves: shape sense, intuition, memory