Is $500/month a typical rate in your area? That is unheard of around me, but I think you're in decent shape if so. Make sure you have enough for the security deposit, it's usually one month's rent.
Other recurring expenses to budget for will be utilities (water, sewer, trash, electricity, maybe gas, internet, and phone - you can estimate from whoever pays these where you're staying now), auto fuel/maintenance/insurance unless you use public transport or bike, and food as others have covered (I think your idea of practicing grocery shopping and cooking is a good start).
Beyond that, have realistic expectations for more variable expenses like clothes, home goods, entertainment, eating out, etc. Be mindful, but don't be afraid to treat yourselves every now and then if you can afford to.
It's also wise to save up an emergency fund that can cover you both for six months. It sounds like a lot, and it is, but think of it more as a goal you can keep contributing to with the money you have leftover each month.
Moving can be expensive, especially the first time when you don't have many of the essentials. Keep an eye out in thrift stores and online marketplaces for things like cookware, dishes, vacuums, and furniture. These are often much cheaper second hand, and you can upgrade down the road when you know you're financially comfortable. Good luck!