Firstly, welcome to Linux Mint, and indeed Linux in general, I hope you learn some very useful skills in the next few weeks. You've already passed the first test - don't hesitate to ask for help. Don't be afraid of your problems, in this case you're still going about it the windows way.
In windows, you often go to a website, download an install file, and run it. In linux we have "repositories" (repos for short) which store the most up to date version of software that is known to work with your system. We then have an installation manager which connects to the repos, and lets you install everything.
The manager for mint is mintInstall, which you'll find in the main menu. You browse for a topic which you want the program to deal with (e.g. mp3, genealogy etc.) (rather than the name of a windows program you used to use), then just click install. If there's anything you can't find here, you could try the more complicated package manager (which in fact is where you'll find hellanzb). Note that you probably wont find exact windows programs, but you will find something that does the same job.
If you find you really can't do without some windows program, then you can try running under wine, which is a program designed to do just that, but its configuration isn't trivial. There are however various sources you can go to for help (in addition to our forums). Stuff I recommend reading (in approximate order):
Linux is not Windows: http://linux.oneandoneis2.org/LNW.htm
Linux Mint 6 User Guide: http://ftp.heanet.ie/pub/linuxmint.com/stable/6/user-guide/english.pdf
Using windows programs under linux:
Alternative apps listing: http://www.linuxalt.com/
Wine tutorial (follow the instructions, and only take guidance from the snapshots): http://www.simplehelp.net/2007/05/18/how-to-install-windows-programs-in-linux-using-wine/
[I was going to suggest wine-doors.org, but it seems they've temporarily removed their repos due to lack of funding]
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