Just like most of us aren't going to be tenured professors when we get out of grad school (so we should think about what else we want to do), most of us won't be doing the same work we do now after graduation. Being a teaching assistant doesn't always mean teaching our own classes - so we might not have that experience until we’re somewhat locked into a teaching career. It would be nice to figure out if we even like teaching sooner rather than later! Writing a journal article or a dissertation differs from writing for the general public.
At the same time, graduate students tend not to make that much. Budget advice for graduate students is everywhere. That’s not as an endorsement of "better budgeting" to solve problems of low income. We're not going to buy a house by skipping the avocado toast. We are, however, fairly likely to work multiple jobs: 44% of 25-34 year old workers have a second job. Grad Student Finances has a series about the additional jobs of graduate students.
To be fair, working a second (or third) job in graduate school is not always practical, nor is it always permitted. Sometimes graduate assistant contracts ban side work, and sometimes departments frown on it. And really, any one job should pay a living wage. Still, many of us will work additional jobs in graduate school. We're also likely to want professional development opportunities beyond those our programs offer, especially if we're looking at career paths other than research and possibly teaching.
So let's kill two birds with one stone: we need money, and we need work that relates to what we're going to do after graduate school. If I'm not going to become a professor, my preferred career options are teaching and writing. My professional development needs are therefore fairly similar to those of the folks who will be professors, honestly. I need to teach, and I need to write.
When I was in the math department, being a teaching assistant meant teaching a class. As a neuroscience student with an assistantship in electrical engineering, it did not mean teaching a class anymore. I'm still teaching math, though - just online, and to a younger audience. If you're interested in teaching, you can consider writing centers, and private tutoring as job options as well. Even if you can't always get a whole class, there are quite a few ways to work in education.
Maybe you want to be a writer. A thesis or dissertation is, after all, a huge writing project. Somewhere along the way, you could discover that writing about your topic is the thing you love, and want to make a career in communicating your field to the public. At that point, it may be worthwhile to look into publishing opportunities that pay. Academic publishing may not pay, but writing is a career for some people and other paid outlets do exist. Get out of the ivory tower with your writing and maybe even get paid.
Perhaps you're interested in working for a non-profit. There are commonalities in grant-writing, no matter who the grant is meant for. That gives graduate students in grant-heavy fields a bit of an in. Non-profit organizations may also run community events, which would make event planning a relevant side gig during graduate school.
Whatever it is you want to do after graduate school, there are probably some aspects of it that aren’t part of your graduate education. Unless you get some outside experience with those aspects, you're not going to be much better with those tasks after graduation than you are now. If you need outside employment, getting a job which incorporates some of those skills can kill two birds with one stone. Considering how pressed for time graduate students tend to be, that’s handy!
Have you worked outside your department in graduate school? Was that work connected to what you’d like to do after you graduate?