There aren't a whole lot of options to choose from compared to when 35mm film was in its heyday. Since you don't know much about film, be aware there are different characteristics between them.
I don't know what is still available for B&W, but T-Max and Tri-X were 2 biggies. T-Max was more contrasty. Tri-X showed a lot of detail within the shadows. Am not familiar with Ilford types so can't help you out there.
Color film also varies. Some were geared more towards portraits so the colors are more muted. Flesh-tones are more accurate. Others might have a lot of vibrant color.
Some film will yield more grain than others. Read reviews to see what they say.
Currently I have a roll of Ektar 100 in my refrigerator (some films do better when stored in a refrigerator), but I haven't yet tried it. Make sure to remove the box from the refrigerator and allow it to warm up to room temperature before opening the plastic canister. Those canisters help prevent condensation from forming on the film. I wanted something with vibrant color for landscapes. What I used to use is no longer made. Take a look at the reviews and see if it is one you want to consider. It is ISO 100 so a tripod might be necessary.
https://www.bhphotovideo.com/c/product/585497-USA/Kodak_6031330_35mm_Ektar_100_Color.html
I also have a roll of 400 film but am not able to read what it is through the canister. The Ektar's canister is in a box. This roll of ISO 400 isn't so can't help you there.
And consider getting a film retriever. Once in a while you might accidentally think your film is loaded properly only to find out it isn't advancing. If you rewind an unshot roll of film it all the way with the leader going back into the canister, you will need a retriever to get the end pulled out (but it's fine to rewind it all the way after shooting the roll). My local camera store has been fantastic with helping me, and when the employee found out his film retriever was missing, I bought one of these for him and one for me. He immediately opened up the package and tried it out. He was able to pull the film out on the first try. Not all retrievers work easily.
https://www.amazon.com/gp/product/B00DN1080U/ref=oh_aui_search_detailpage?ie=UTF8&psc=1
Be sure to set your ISO/ASA and don't change it mid-roll. That is one big difference between film and digital. You need to shoot an entire roll at one consistent speed. There is something called push processing that will allow a film of one ISO/ASA be shot at a different ISO/ASA, but I've never done that myself. Sometimes people needed a faster ISO than the film was made for. Read up on it, and if you try it out, BE SURE to let the lab know the film was shot at a different ISO. If they don't know, they will process it normally, and you might lose your photos.