In addition to what Bob said, since you clearly have access to a working system, just download something like Firefox from there and put it onto a flash drive or a CD. Then install it on the other system.
Finally, since you ran into trouble within a couple of days, I'll toss in my tips for keeping a smooth running system.
TIPS FOR A SMOOTH RUNNING SYSTEM
The more of these suggestions you follow, the fewer problems you should have. They won't solve any existing problems you have, but if you follow them all you should be able to avoid virtually all problems in the future.
Things you should NOT do
1: Use Internet Explorer (1)
2: Use any browser based on Internet Explorer
3: Use Outlook or Outlook Express (2)
4: Open email attachments you haven't manually scanned with your virus scanner
5: Open email attachments you were not expecting, no matter who they appear to be from
6: Respond to spam messages, including using unsubscribe links
7: Visit questionable websites (e.g. porn, warez, hacking)
8: Poke unnecessary holes in your firewall by clicking "Allow" every time some program requests access to the Internet (3)
9: Click directly on links in email messages
10: Use file sharing or P2P programs
11: Use pirated programs
Things you SHOULD do
1: Use a non-IE or IE based browser (4)
2: Always have an up to date virus scanner running (5)
3: Always have a firewall running (6)
4: Install all the latest security updates (7)(8)(9)
5: Delete all unsolicited emails containing attachments without reading
6: Manually scan all email attachments with your virus scanner, regardless of whether it's supposed to be done automatically
7: Copy and paste URLs from email messages into your web browser
8: Inspect links copied and pasted into your web browser to ensure they don't seem to contain a second/different address
9: Establish a regular backup regimin (10)(11)
10: Make regular checks of your backup media to ensure it is still good (12)
(1) Sadly sometimes this is unavoidable, so only use IE when the site absolutely will not work with any other browser and you cannot get that information/service anywhere else, and only use IE for that one specific site.
(2) Outlook and Outlook Express are very insecure, and basically invite spam. Possible replacements include Mozilla Thunderbird, Eudora, The Bat, and dozens of others.
(3) When it doubt over whether or not to allow some program, use Google to find out what it is and whether or not it needs access to the Internet. Otherwise, denying access is the safest course of action, since you can always change the rule later.
(4) On Windows your options include: Mozilla Firefox, Seamonkey, Opera, Flock, and Safari. It doesn't matter which one you pick so much as that you pick one of them and use it over IE.
(5) AVG Free and Avast are available if you need a decent free virus scanner
(6) XP/Vista's firewall is probably good enough for 99% of all Windows users, but other options include ZoneAlarm, Outpost Firewall, and Comodo. If you have a router with a firewall built into it, there is no need for any of the aforementioned firewalls to be running.
(7) Microsoft's usual system is to release security updates every second Tuesday of the month.
(8) Use of Windows Update on Windows operating systems prior to Windows Vista requires Internet Explorer, and is thus a valid exception to the No IE rule.
(9) Service packs should ALWAYS be installed. They frequently contain security updates that will ONLY be found in that service pack.
(10) You can go with a full fledged backup program, or simply copying important files onto a CD/DVD/Flash drive.
(11) I'd recommend a tiered backup system. For example, you might have 5 rewritable DVDs, and every day you burn your backup onto a new disc. On the 6th day, you erase the disc for Day #1 for your backup, and so on so that you have multiple backups should one disc ever go bad.
(12) Replace rewritable CDs and DVDs approximately every 3-6 months.