To clarify: do you mean literally grad students while they're still students, or after they graduate?
While still in school, the most common (by far) are teaching assistantships (TAs) and research assistantships (RAs). Normally you start out as a TA for a couple of years, and then when you decide on a subfield and find an advisor, you work for him or her as an RA (ideally on their own funding, but in my case I had to apply for my own grant, which fortunately succeeded). While you're still taking classes (which typically lasts two years or just slightly more for PhD students), your summers are likely to be open (and not TA-supported), so summer internships at NASA sites, various observatories, and national laboratories are good options. (https://aas.org/careers/internships-summer-jobs is a great resource, and I'd bet aps.org has something similar.) I spent a summer at Fermilab as an undergrad (helping to build a drift chamber for an experiment) and another at Argonne as a grad student (helping write simulation code, I think related to lattice gauge theory). I also spent a summer in grad school at Bell Labs doing calculations for a physical simulator (a specific type of test equipment for electronic phone switches). Bell Labs is probably less of an option for physics/astrophysics students these days, but the commercial space industry is a much bigger one; see, for example, this article that just showed up a few minutes ago: https://www.geekwire.com/2018/brooke-owens-women-aerospace/
Once you graduate (and I'll assume you have a PhD, since that's the most common option), the traditional path is to apply for postdoc positions, and again, the AAS has a bunch of listings: https://jobregister.aas.org/ . You should potentially be prepared to go overseas for some of them; I recall some interesting ones in the Netherlands and I think Australia when I was looking, as well as Canada and the US. But you may also choose to go a less traditional route, particularly if you have a strong interest in instrumentation or computer science/data science. I'm not completely sure of this, but I believe there are staff (support) positions associated with most observatories and initiatives (like the Dark Energy Survey, DES), both in the telescope- and instrument-building area and definitely in the data-sciences area. The current generation of instruments are generating data at very high rates, and the next generation will do so at ridiculous rates; writing software to capture/store/transmit all of the data, to validate it (data quality), and to search and process it is going to require a huge effort. I work with big data of a non-scientific nature; our team sizes are not small, and the upcoming generation of instruments will dwarf our data rates.
And as my own case shows, you can end up in an entirely non-astrophysics career if you so desire. A background in physics and astrophysics tends to provide a very good foundation for a lot of other types of work, although it's certainly not a slam-dunk: you'll need to be able to adapt, learn new things on your own, collaborate in different ways from what's typical in academia, potentially break out of "research mode" and get into a more results-driven/engineering mindset, etc.
This professional recommends the following next steps:
- Check out the internship and postdoc links above for an overview of the kinds of positions available on the traditional, academic track.