Attempt the problem on your own first.
Use outside resources and your own research to try and answer the question.
If you get stuck and can't do it on your own without a little guidance, use outside resources to read about the problem, look through other examples to get an idea of the general procedure, and get a better grasp of what the final answer should actually look like.
Personally, I don't like asking a professor for help or going to my TA's office hours unless I absolutely cannot figure out the problem on my own. So the next step might deviate from other people's opinions on how to use solutions manuals.
- Use the solution manual as a quick hint as to the next step.
If you make it through some of the problem and get stuck somewhere, and you cannot find any kind of answer online or in the textbook, then use the solution manual to give you a quick hint as to how you should proceed. It's sometimes useful to see how the book author approached an integration or some weird algebraic step. If you make it to step three, then step four is the critical part of this method.
- Use the hint you just took as an opportunity to further your understanding. Work through similar examples outside of the actual homework problem (in the case of a math problem), use a keyword in the solution to read about that step (in the case of a physics problem), etc. Do further research on that particular step, so that when you encounter a similar problem in the future, you will know exactly what to do.
Don't just write down the solution and move on, learn from it to deepen your understanding of the problem. After using the solutions manual and not making any progress on understanding the problem, or that particular step, then I use the following step.
- Go to a professor or your TA's office hours to ask the question and get a one-on-one dialogue going about your misconception, why your attempt failed, and/or why the author of the solution did that particular step.
For a mathematics question, I typically ask them on math.stackexchange.com first, as I usually get a great response within minutes of my post. For a physics or other technical/theoretical questions, I found it best to discuss them with someone.
By going through the steps as I have listed them above, I have found that by the time I go to a professor or TA, I can explain the issue I'm having and convey it without stumbling over myself. It taught me to figure out the issue I'm having exactly and to be able to ask a very specific question, rather than going to a professor and asking some vague question about an assigned homework problem and looking for them to give me the solution.
Apologies for the novel, I hope this helps.