Go with 400. Straight up.
Do not worry about overexposing with it... film these days handles overexposure extremely well, and if youre using 35mm, there's very little to worry about when it comes to additional grain characteristics that might present themselves, because, well.. youre pretty much doomed to always have them unless youre shooting very specific stuff that won't be as good as a general use film.
I use a Nikon L35AF as my point and shoot camera in snow conditions, which has a fairly fast lens, but if you get something with a zoom lens, fully expect that thing to have a pretty slow lens - like a max of F5.6 or therabouts.. so you'll need at least 400 speed anyway, and might even want to push that stuff to 800 and compensate with a longer development time.
I use Tmax 400 as my go-to 35mm stock (unless I can't get it for a good price, in which case I use ilford HP5 which is also amazing). Some other great black and white films that generally run a little cheaper are Fomapan 400 (Arista EDU 400) and Kentmere 400 (Ilford Pan 400) and I have an affinity for each of these as well.
and if I'm shooting colour, it's generally always Lomo 400 or 800 since Portra or Ektar are priced pretty brutally these days.
Both Lomo 400 and 800 are films that were originally made for Kodak disposable cameras, and are pretty simple in their emulsion formula - with a decent amount of grain and some white balance shift that's not quite true to real life... and I overexpose the shit out of them and they render some amazing photos in these cases. Theyre cheaper than the Pro Kodak films but still have a ton of dynamic range when you overexpose them...