Sounds like you have a DSL service (as do I, at the moment).
Let me first suggest something: do you really need to replace the modem/router? A new one probably won't be any "Faster" as you are likely throttled by your ISP (ATT? Verizon? Joe's Telecom?).
If you have a working modem and router, just buy your new computer, plug in, (or configure wireless to match the router's security settings) and probably you are good to go. It doesn't sound like you have any exotic needs like superfast gaming, gigabit media distribution, or any of that.
If for some reason you do need to replace your modem & router, I actually recommend against getting a combo. I've had lots of friends experience annoyances with them. I actually purposefully maintain them as separate so that:
1. I find it easier to troubleshoot the rare problem when they are separate, because it's easier (at least for me) to tell if the problem is with internet connectivity or a router hiccup.
2. I toy with going back to a cable modem instead of DSL; while most combo devices allow you to turn off the DSL modem piece, they still are really optimized for DSL.
3. At this point, I never recommend anyone buy anything other than a gigabit router, and it's cheaper to buy a separate gigabit router than buy a combo device that can handle gigabit speeds. Essentially, routers have reached the limit of what is needed and the router you buy today should last you 5+ years. My router is 2 years old, and there's nothing on the horizon that suggests it will reach obsolescence within the next decade. Lightning will probably zap it first. Media streaming (over the internet and within the home) is the new TV, and it's a lot better to do it over a gigabit network than over 100baseT, as you have bandwidth to spare then.
That all said, I run an ActionTec G701D DSL modem and a Netgear Gigabit N+ router and have had no serious issues with either. The router is a little slow to reboot (takes about 2 minutes to finish the whole connectivity cycle), but this only matters if you are troubleshooting some weird issue.
Good luck!
MArc