Yo Quacker! You have gotten some good advice, but the real question hasn't been really addressed. Can you use an XLR to 3.5mm cable and plug a low impedence dynamic mic into your soundcard and get good results? The basic answer is-No. Computer soundcards are not designed to process the low impedence, low output of real microphones. If you do that, you will hear something, but it will be real low volume, with lots of noise, and will suck in general. The second problem is that computer soundcards are not designed for processing critical audio, and usually cost about $5 to produce.
There are 3 common solutions to this problem, and there are some subvariations of each of them. I'll show you some cheap examples of each.
The first is to bypass the computer completely, using a standalone digital recorder. Most, but not all of these can then upload the audio data, usually as a WAV. file, to a real computer for further processing, and/or conversion to MP3/CD or whatever. Example:
http://www.8thstreet.com/product.asp?ProductCode=44756&Category=Recorders
The second is to replace the soundcard in the computer with one that *is* designed for processing critical audio, and use an external microphone preamplifer to raise the mic output to line level. Example:
http://www.8thstreet.com/product.asp?ProductCode=3598&Category=Audio_Interfaces
and
http://www.8thstreet.com/product.asp?ProductCode=10658&Category=Audio_Processors
This second solution above has largely been outmoded/abandoned and replaced by the third, most common (currently) approach, which is to use an audio interface, which combines one or more mic preamps with an analog to digital covertor. These transfer the data, usually by USB or firewire, directly to the computer, and sometimes come with some kind of audio processing software. Generally, but not always, USB is used to transfer 1 or 2 channels simultaneously, and firewire for larger numbers of signals. Many such interfaces are also powered by the USB, so they can work in the field with a laptop. Example:
http://www.8thstreet.com/product.asp?ProductCode=27765&Category=Audio_Interfaces
That one is cheap, but doesn't produce phantom power, so it can only be used with dynamic mics. It doesn't come with software, but you will see that there is free basic audio software available, such as Audacity, or the more versatile Reaper for $50 or so.
This one, which was mentioned above, is more versatile, with 2 mic inputs and phantom power. You may decide down the road that you'd like to try stereo recording or condenser mics, and I believe that 2 mic inputs with phantom power is the most basic entry level setup. Anything less than that involves too many limitations:
http://www.8thstreet.com/product.asp?ProductCode=12031&Category=Audio_Interfaces
Here's another inexpensive unit, which, although specialized for guitars, would work for you and has very good sound quality for the price range:
http://www.8thstreet.com/product.asp?ProductCode=60669&Category=Audio_Interfaces
A subcategory of this is to use a USB mic or an inline USB convertor. This is either a mic with built in preamp and analog to digital converter (usually a condenser, not a dynamic), such as:
http://www.8thstreet.com/product.asp?ProductCode=41780&Category=Microphones
or an inline preamp and A-D converter which converts the output of a standard mic (including dynamics) to a digital output by USB. These may or may not produce phantom power, and usually don't offer headphone outputs. The preamps are small, and generally not the best. In short, they work, and could be an option if money is tight. but lack versatility. Example:
http://www.8thstreet.com/product.asp?ProductCode=42882&Category=Audio_Interfaces
OK, all that said, a few words on mics:
For quite a few years, the Shure SM57 has been the cheap go-to dynamic mic in the USA, especially for snare drum and guitar amps, where a cheap mic is often the best choice. Many great engineers use it and swear by it. There are, however, a ton of perfectly good dynamic stage mics that make excellent substitutes. Lots of folks will disagree with me on this, but I believe the SM57, while a perfectly good cheap dynamic, is nothing special. In Europe, the Austrian made AKG and German made Sennheiser mics have been used extensively by everybody from the Beatles to the Grateful Dead (OK-not European, but they know a good mic when they hear one). Often these mics can be found used on ebay or Craigslistat a lower price, because not as many Americans are familiar with them. For brass, I like AKG D320, D690, D770, and Sennheiser e835. For my money, the D320 is the oldest, cheapest, and best of them. Sennheiser MD421 is one of the best brass mics on the planet, but is generally out of your price range, unless you get lucky at a pawn shop or something. Honorable mention goes to the dirt-cheap Behringer XM8500-($20) brand new. Many people will jump on me for recommending *anything* made by Behringer, as they are a company known for producing cheap reverse-engineered copies of other people's gear, and for spotty quality control and questionable reliability. The XM8500 appears to be more a knockoff of an AKG than a Shure, but I have to admit- it is a robustly built, good sounding handheld dynamic mic that would work fine on brass, and save you a bunch of money to put toward your interface, mic stand, cables, etc. Like I said, folks may jump on me for recommending it, but if they do, ask them, "Have you ever used an XM8500?", and I'm betting the answer will be-no. Everybody I know of that has used one agrees it's a pretty good mic:
http://www.8thstreet.com/product.asp?ProductCode=6535&Category=Microphones
You will notice that ribbon mics were mentioned above. These are a special subcategory of dynamic mics that use a thin metal ribbon instead of a diaphragm. They have been used on brass for many years with considerable success. They come mostly in 2 categories- wicked expensive and to die for (AEA, Royer, RCA, etc.) and much more recent and far cheaper Chinese mics (Shinybox, Nady, and a host of other brand names stuck on the same mics). Ribbons are a little fragile, and need to be handled with care. They are not mics I generally recommend to beginners, but may turn out to be very good for recording your trombone. Consider a ribbon down the road.
Hope this helps. Best of luck-Richie