I am an astrophysicist/astronomer, and I can vouch for that as a field in which amateurs still play a major role. Hell, my first paper on which I was the first author was something that anyone with a decent set of programming skills and a mild knowledge of astronomy could do - the trick was knowing that it needed to be done. As far as how to get involved, I have several suggestions.
1. Go back to school, formally or informally. If you have a university with an astronomy or physics department, get involved there, perhaps by taking a class or two part time. This serves a threefold purpose. The most obvious is that it is a great opportunity to learn the basics. Second, it will give you access to university resources, such as library (including journals and databases online!) access, computer access, and sometimes telescope access. Finally, it will give you access to professionals. Many professors and researchers are in need of people to do their side project. These often get done by undergrads, and if you go back to school you can get involved. All you do is ask around the department if anyone needs free help, and then find someone you want to work with. You might get a coauthorship out of it. This is how I did my first paper as an undergrad.
2. Look online for topics that amateurs with a telescope can help with. Try variable stars
, asteroids, comets, supernovae, gamma ray bursts, etc. Alternatively, if you are into programming, CS, or data methods and their applications to astronomy, this is a great area to make contributions with little-to-no startup cost. Rapid time series analysis and signal processing are big these days, among many other things.
3. Don't be ignorant. Read introductory textbooks. Refresh yourself on math, physics, and programming. Read wikipedia. Read arxiv astro-ph
and use ADS
There are other non-astro things I can think of too. I'm into paleontology as a hobby - its another field that amateurs routinely make contributions in, but that's decreasing with time. Fossil collecting can be really fun while also getting you outdoors. Unfortunately, this is highly dependent on your location, local laws, and other factors out of your control. Same with amateur archaeology - but this is even harder to do. In some parts of the US, for example, amateurs have found major native american archaeological sites, which they then call in professionals to help with. Another way to get involved in these fields is to volunteer at a natural history museum or a university, similar to how I described for astronomy, but it is usually easier because they need people to help on digs and things. There is less potential for a publication, though. Computer Science, mathematics, and software engineering also seem like fields an amateur could make contributions in.