Ohheck, how are ya? If your just going to surf the net and check e-mail you do not need a new mac by any means. Honestly, IMO (in my opinion), get a used one. You don't have a need for a 1.83 GHz core duo. For what your going to use it for... its just plain' ol' over kill. I would hate to see you waste your money on something you don't need.
Also, how fast the internet moves is primarirly based on how fast your internet connection is not the pc itself.
First what you have to decide on is whether or not you want a laptop (which I think is your best bet) or a desktop. The average price is about $600-$800... about half of a new mac.
LAPTOP
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If you go for a laptop you'll enjoy the benfits of bringing it where ever you want (deck, front of the house, bathroom, bedroom, etc.). Its not big an clunky. You won't have to buy an extra mouse, keyboard, or monitor). Many are equiped with a wireless card already (so you can go to places like barns and noble etc. and connect to the internet or if you have wireless at home). I would look on a place like
www.techrestore.com or
www.ebay.com A few good examples would be
http://techrestore.com/xcart/product.php?productid=17100&cat=254&page=1
http://cgi.ebay.com/Apple-ibook-G4-...76QQihZ005QQcategoryZ4602QQrdZ1QQcmdZViewItem
http://cgi.ebay.com/Apple-ibook-1-3...38QQihZ002QQcategoryZ4602QQrdZ1QQcmdZViewItem
http://cgi.ebay.com/1GHz-15-Apple-P...1QQihZ001QQcategoryZ14909QQrdZ1QQcmdZViewItem
Desktop
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Desktops are what your probally used to. Since there is an actual monitor, things are pretty easy to see (bigger). Everything isn't compact. They are much easier to upgrade and maintain. Not to mention they are normally cheaper then laptops. Here, the iMac is a good option, for it already comes w/ a monitor. Though, you can use your current monitor w/ the correct cable. Here are some good examples...
http://cgi.ebay.com/Apple-17-iMac-G...ryZ80040QQssPageNameZWDVWQQrdZ1QQcmdZViewItem
http://cgi.ebay.com/Apple-Imac-G4-1...ryZ80040QQssPageNameZWDVWQQrdZ1QQcmdZViewItem
http://cgi.ebay.com/APPLE-MAC-MINI-...4QQihZ016QQcategoryZ86702QQrdZ1QQcmdZViewItem (this is pretty sweet, comes w/ a warrenty and all)
http://cgi.ebay.com/Apple-Mac-MIni-...QQihZ004QQcategoryZ114190QQrdZ1QQcmdZViewItem
I really do hope this helps and hope that you go this route instead of buying new. The G4 processor is a great processor and can easily keep up with what you want to do. If you have any questions please let me know. I have been dealing w/ computers since I was seven and currenty work full time in IT.