Do I just have to spend hours trawling the site to find that one unanswered question that I happen to know the answer to?
This may be one of the biggest issues new users face on any site on the Stack Exchange network: You want to contribute in a meaningful way, but are restricted until you've contributed in a meaningful way. There's no easy out of this conundrum as it is a network-wide initiative that attempts to set a boundary between noise and quality.
We are a friendly bunch, so join in when you can.
Here are some suggestions that might help one gain reputation:
Post quality answers. This should be numbered 1, 2 and 3. But a good answer will more often than not trump a fast answer. Though there are cases where it does not.
Setup a good but short list of Interesting and Ignored tags. For example, one may have macros, tikz-pgf and tex-core as interesting tags. This will help you see questions, which you can answer, quicker.
Avoid Wall of Text questions. They take way too much effort for little reward. And usually are syntax errors or bad structure.
Post an answer even though the question has 1–2 or even 3 answers. In these cases, take your time and answer well. This will usually net you a good sum of rep.
Learn when to edit. Post a short answer at first and then edit. You have <5 minutes to make that answer shine.
Be humble, thorough and fair. There are a lot of smart people out there and many will know much more than you about the subject. Be thorough in the code you post, check it for syntax errors and make sure it fits the question. And if you see that the correct answer is already there, upvote it, that person deserves the rep.
I noticed that you have accounts on other sites in the Stack Exchange network, some of which have higher reputation than your TeX.SE account. Once any of your accounts reaches 200 reputation you will receive an association bonus of 100 reputation on all your accounts.
You can also suggest edits to other users' posts -- you will gain 2 reputation for each suggested edit that is approved by users with higher reputation. You should edit posts to correct spelling or grammar mistakes, fix code or quote formatting, etc.
I would also suggest to ask (good) questions.
Of course, some questions are more 'trendy' and thus more likely to get dozen of up-votes.
But an interesting and well asked question is likely to be rewarded by a few up-votes, what is enough to reach the 15 rep cap.
Interesting means a question that matters (if it really matters to you, it matters for us and it is not too specific or spam!) and have not been already answered (no: all questions have not been already asked!).
Well asked means clear and concise, well formatted (using code blocks, text formatting), with a MWE (if needed), and tagged appropriately.
Moreover, this adds to the quality/interest of the site... and it's the first step in this virtuous circle: learning LaTeX -> enhanced LaTeX skills -> more knowledge to share... and more will to learn new areas!