Honestly there is no SPECIFIC sole model. As your needs are 'basic' 90% consumer needs (no games, video editing, homework isn't crunching 1000x1000 row/column spreadsheet, etc.) there is ALOT out there to pick from based on your personal choice, running from $199 to $300 for Laptops, All-in-Ones, and Desktops. The Laptops sell the most, and any of them will meet those needs, BUT NO COMPUTER WILL SPEED UP YOUR INTERNET.
If your issue is things are 'slow', and those 'things' are all Internet based, that can be any number of other things, like for example your Mom/Dad was cheap and didnt' get a Internet fast enough for a entire family of computers. Or can be Mom/Dad never 'got the new router' they offered and your old one sucks. Or there is TOO MANY devices connected, or your using the slowest 'wireless' signal (i.e. 802.11b or 802.11g instead of N or AC). Etc. Etc. Etc.
You can check on Amazon, Walmart, Best buy, Staples, etc. there is plenty to be had. Check the features, feel the systems. find one that meets your feel. Just price shop and get the best deal for the hardware and features one has over another, most is basic (i.e. one computer had 2GB of RAM another has 8GB, well 8 > 2 so... ).