First of all, don't worry about your rating. Particularly beginners will have large fluctuations in rating and in the end who cares about that number anyway?
Regarding tactics training, doing online puzzles is a very good way. Many websites offer these for free. I like the ones on lichess, because they are taken from real games and therefore tend to be a bit more realistic/messy than composed problems. However as a beginner you probably prefer cleaner problems focusing on just one or two motifs. But in any case, any good website will present you problem close to your level so that should not be a problem.
When you solve these puzzles (or fail to solve), I suggest to not stop there, but analyze each puzzle a bit more.
Particularly you should figure out which tactical motifs are relevant in this puzzle. Chesstempo can help you here, as once you solve (or fail to solve) a problem it will show you the motifs as "tags". Make sure you can see all of the motifs in the solution.
Another thing you can try is to take the starting position and remove all pieces from the board that are not necessary for the problem. This will give you a cleaner/clearer image of the tactics.
The aim of tactic training is to store certain patterns (positions of relevant pieces) in your brain and associate them with a tactic. You only store the relevant pieces, not the complete position, because this way the patterns are re-usable for many situations. As an example, a simple pattern could be: two pieces on the same row, two squares apart (e.g. on d5 and b5). This pattern would be associated with a "pawn fork".
When you solve puzzles later, you can quickly check whether the problem matches any of the stored patterns. Often it will match more than one, giving you more than one candidate move.
At this point it is time for concrete calculations in order to find the best move among the candidate moves and also to make sure that the tactics works (that there are no defenses). This involves for instance checking whether there are any additional pieces (compared to your stored pattern) that defend essential squares or whether the opponent has intermediate checks, etc.
In summary my suggestion is to:
- do tactics puzzles (online or on computer or from books) and analyze them
- try to find and name all the tactical motifs in the puzzle
- remove pieces from the problem to get a clearer picture
- memorize the relevant piece positions
- re-check whether the tactic works