As far as connecting to the internet, it depends a bit on what equipment (eg router) you have at present.
There are still some people with USB modems for their internet access, and I use a mobile network 'dongle' at home.
If you are using USB then you may need to find the original CD that came with it, but the likelihood is that you currently use a router.
On the basis that many people have a normal phone line (ie not Virgin Media cable) then some routers can have many PCs plugged in, or using the connection simultaneously (it's their purpose, to connect one or many 'local' machines in your home, to a single connection at the ISP, and allow your phone line to carry data for as many PCs as you have active at the time.
Some routers have a single ethernet (RJ45) socket for a PC to plug in, many have 4 ports, and newer routers (especially those supplied by ISPs) may have both wired (RJ45 cables) and wireless connections.
Many laptops these days include a wireless card, so they can connect to any nearby network (and each network ought to be password protected - it's a way to reduce the chance of 'snooping' on what's being sent and received).
As you plan to use MS Office, and have not mentioned any Apple kit, then I would suggest you look for a machine running Windows 7.
The minimum spec for a Windows 7 machine is said to be one with 2 GB of RAM.
You could now do a search on Froogle (Google here and then it comes down to how large a screen you want to have.
I've recently bought a Compaq CQ61 which was just under 250 pounds in Asda Living (but unfortunately it seems like they were running down stock so reduced them by 100 quid to clear the shelves), and that was for a machine with 15.6" widescreen, 2 GB RAM and a generous hard drive (160 GB).
If you can find some system with a screen of 14" or more, at least 2 GB RAM (4 GB would be nice) and under 325 then I'd think it would be a pretty good deal (within budget and allows for extras you might buy, like a wireless router to make it usable anywhere in the house). Hope that gets you started...
Re MS Office - plenty of people on MN have youngsters in their family so may be eligible for an educational discount on MS Office, making it more affordable.
Alternatively, you could try out OpenOffice (a free alternative, which has recorded over 100M downloads) and see if it will open your existing documents and save versions you can open - if that's the case, it could save you the cost of the Microsoft version altogether.