Hi Nick,
This is a really good question.
There are any number of ways to go about this, but the most logical place to start is with your hosting company. Do they have a built-in formmail script that you can use? (If you use a cPanel host, the answer to this question is "yes.") If they have it built-in, all you have to do is create a web page with the form fields, and add the variables in your HTML coding that call the script. Your host will be able to provide you with instructions on how to use it. It's really pretty much copy/paste. If you can build a basic web page, you can do a contact form.
If your hosting company doesn't offer this, you should be able to upload and use a formmail script on your hosting account. I recommend
NMS Formmail. It is free, secure, and works like a champ.
I recommend
against using anything
Matt has written (e.g.,
Matt's Formmail) for security reasons. Every version before the current has been exploited, and among security experts it's considered a only a matter of time before the current one is cracked.
Alternatively you can use a third-party formmail service. There are both free and paid, although paying for it doesn't buy you any advantage that you can't also get from a free solution. Paying for it does not buy you any extra advantages, guarantees or assurances that you can't get from free. (In some cases paying is better. This is not one of them.)
Third-party solutions make me nervous, because no matter what the site's policies profess, you simply have no guarantees that their system is secure. For all any of us know, they could have an employee who's been bought off by spammers who has quietly added a few little lines into their scripts which forward your customers' confidential information to a database of some type, or directly to spammers. Every formmail provider on the planet is going to swear up and down that this isn't the case, but the fact is that none of us run their servers, nor have direct access to their scripts, to know this for sure. Maybe if I hadn't seen worst-case scenarios in action myself in the past, I wouldn't be so paranoid about this, but it doesn't take much to get just one employee to stray and do things behind-the-scenes without anybody else knowing it.
Many hosts offer a pre-installed solution, or a click-to-install solution, all of which work great. It's also not very hard to install and configure a formmail script. Such solutions are accessible enough that IMO, I would opt for one of those in a heartbeat over using a third-party installation.
If you have any questions, please LMK!

Bailey