I fully support buying your own modem and keeping the modem & router separate. I feel like the all-in-ones seem to crap out a LOT more frequent than separate devices, but maybe that's just my limited experience. I bought a DOCSIS 3 motorola modem for $50 from Amazon. I got a fancy router that is better than 99% of people need. Total cost was under $200. I could easily get both for under $100. Make sure the modem you buy is on the supported list for your carrier. Google "CompanyName modem compatibility" and you should get an official document from them with what models they currently support. You may be able to get something cheaper that will work, but if you have any issues, they will blame it on your modem.
On a side note, not every ISP will allow you to buy your own modem. If you have VOIP, they will often force you to use their rental, even though there are VOIP modems you can purchase. ATT UVerse would not allow me to buy a modem. I had signed up for TV and Internet initially, and they will not allow TV & Internet users to buy a modem. I did it to avoid a $150 install charge and I got a $300 gift card for keeping TV for 2 months. To buy my own modem, I would have had to cancel service, wait 30 days, and then resign up with a dry-loop internet account. They would then charge me $150 for an install that was already installed. I decided to just go to cable. I highly recommend that if you are looking at buying a modem, make sure you price shop the competition, as this might be a good time to switch providers.