As for what patterns, there are a lot of those, too many to mention.
What is the goal? The goal is to find the best move, though I know that is pretty broad.
You have around 600 rating in puzzles. Those puzzles are in most cases straightforward. It will be either 1 or 2 moves checkmate, or some easier 2-3 move combination.
You should do it like this. Take your time, try not to guess the solution. Look at the position. Count the material. Is it equal or not? If for instance you are down a rook, then a simple bishop capture will not be the solution most likely, as you would still be losing the game, so you either have to find a bigger gain, or in many cases, checkmate (sometimes even stuff like perpetual check to save a draw if there is nothing better). Then see how many times the opponents and your pieces are defended and how many times those are attacked. It might be useful.
Now you've assessed the situation, look at the forcing moves first - checks/checkmates, captures and attacks. See if any of those work to your advantage. Don't be trigger happy to take a bishop if you are missing a checkmate. When you find a solution, double check it to make sure that opponent doesn't have some greater threat. For instance you might capture a rook, but be checkmated on the next move. If that is the case, you have too look further.
By practicing every day in this manner, you'll gradually get better at it.
What exactly are the goals for Puzzles?
I'm not sure if I'm satisfied with that answer (I have begun to slow down to think about moves) but thank you.
I would like to give you a better answer, believe me. 
It all comes down to practice and hard work, you can't really force an improvement. Some tactical books will surely be great as well, as there you would be able to learn by motifs/patterns and then practice it all online too. But I haven't done that myself.
Check out the topic selection. Go to 'Puzzles' and then 'Custom'. Here you can select what kind of topics would you like to practice. If you select a topic, you will get puzzles of that topic (e.g. Mate in 1 or Vulnerable King).
At beginner level, I noticed that most puzzles are about Mate in 1, Mate in 2, or capturing the Queen or Rook, or finding a fork. Look for patternsof this kind.
- Can I checkmate the king on the back rank? (If the king is sitting behind 3 pawns, then the answer is highly likely to be yes, but if you could checkmate with several pieces, check which one can be moved without being captured)
- Can I checkmate by moving the Queen to a square protected by a Bishop?
Etc. These are very common solutions in lower rated puzzles. After a while, you'll learn to recognise these patterns, and find them easily. Then you'll get more difficult puzzles. I'm still trying to work out what kind of solutions should I look for around 800 rated puzzles, but eventually I'll find out.
But I really recommend selecting a topic. I did it accidentally, and got a bit annoyed by having to do the same thing over and over again, without learning anything new. Then I found that I had some topics ticked. At least I know those ones quite well now.
You should not rush through the puzzles. Before you should start a new type of puzzle, study the motif associated with it! (Trapping a piece, sacrifices, etc.)
Hi! My name is Lauren Goodkind and I’m a chess coach and chess book author based in California.
Puzzles are great to help you recognize specific tactic patterns in actual games.
I offer beginner interactive puzzles on my website: https://www.chessbylauren.com/two-choice-puzzles.php
I’m sure these puzzles will benefit you since you have to figure out which piece to capture.
Here’s another suggestion to help you get better: consider all checks and captures. Also, try to find a winning and forcing line as you play games.
I hope that this helps.
This one's a broader question. I don't even know the answer I'm looking for. I don't even know if the topic is the correct question to ask.
I ask this because I understand that some puzzles are looking for Checks or Checkmates, but then there's a few Puzzles in the mix with "odd" solutions. Some puzzles require capturing an arbitrary piece on the board. Some puzzles require placing a specific piece on a specific spot. Sometimes, these are all steps leading to a solution that I would have never found without the Hint feature.
So, I think the better question would be what patterns or moves should I try to recognize in Puzzles?
I wish I could visualize what I'm describing but the diagram feature is just as strange to use.