Firstly, think about where you'll be putting the website. (i.e.which provider will host it). At one time most ISPs offered free web-hosting to all their customers but far fewer do so now. Anyway, the downside to using such web-hosting is that the domain name usually includes the customer's name in the URL. (For example, if you've got an account under the name of Ian Brilliant with an ISP called Sample ISP, your free web space would be at somewhere like www.sampleisp.co.uk/users/i/ianbrilliant. Not everybody wants their own name to appear in the URL of their website).
If you're not going to get your website hosted (for free) by your ISP then you need to be looking for free hosting, together with a 'proper' web address, such as ianshobby.co.uk. To do that you have to pay for a domain name but it only costs a few pounds. (Remember though that you don't really 'buy' a domain name. In effect, you only rent it because you have to periodically renew your right to use the name). If you plan to follow that route, it makes sense to use a web-hosting service that offers free web space in return for getting your domain name through them, such as Freeola:
http://freeola.com/free-web-site/
Alternatively you avoid the small cost of registering (and retaining) a domain name by using a service such as that offered by Freewebspace.com:
http://www.freewebspace.com/
Remember though that such services usually make their money out of placing adverts onto free web sites, which not everybody wants.
Once you've decided where you site will be hosted (and whether you want to register a domain name), you'll need to design your website. You can do it using nothing more complex than a word processor but it makes more sense to use proper software for the job. This would probably meet your needs:
http://www.serif.com/web-design-software/?mc=FSSWEBPLUS
As far as getting the videos onto your website is concerned, most people find it easiest to simply put them onto Youtube and then include the embed code into their own website, rather than actually putting the videos on the website itself.
As a final thought, I'll suggest an alternative to having a normal website, which is to use a blog. Services like Blogger.com (aka Blogspot.com) are easy to use and it's a simple task to embed Youtube videos into your blog pages.