Gratz on the birthday present. For river tables, the primary use of the router will be as a slab flattening jig. Ideally, a bottom clearing bit in 8mm shank, but 1/4 will do if you're in NA. Size will just be smaller and clearing passes more just to avoid shank deflection.
The following YouTube video might be useful to understand what is required. Typically though as Paul mentioned, river tables are constructed differently than the video which is uses the router to fake the live edge. Given that most RTs are pretty opaque, this might be an option if you don't care to see depth.
Welcome and enjoy.
Edit: appendix - for plunging and hogging out the river, a spiral upcut might be useful to clear the copious amounts of chips. You can get away with a cheaper straight plunge bit, the difference being the start will be slower but once the channel is made, not much. In case you're new to routers, avoid straight bits that aren't plunge rated as you can only start from the outside and cost saving isn't much.