Difference between Self-Service, Customer, and Partner portals?

As mentioned, these have all been subsumed into Community, the portal to rule them all.

For the curious, here's the lineage.

Self-Service

This was the very first portal. It's main draw is it's FREE. User management is funky (not stored in the User table), and customization is extremely limited.

Partner / Customer Portal

These are the second portal varieties. They're effectively the same in terms of customization and user management, but just targeted at different user types. Partner portal is for your, well, Partners, who need to work with Leads, Contacts, Opportunities, etc. Customer Portal is more focused on support needs, so your customers can interact with cases. Partner licenses, having more access to the core CRM features, were significantly more expensive.

Community

I haven't gotten much chance to try these out, but the high level idea is you no longer have to worry about all the different portal types there is just one.

I'd discourage Portal creation other than in Dev orgs - any new Portal developmnent should stand on the shoulders of Communities instead. If you replace Portal with Community, a lot of the existing documentation holds true.

Also watch out, I don't think you will get a Portal or a Community to live on your Salesforce domain.

Which one of these portals would be the best for what I'm trying to do?

If you must use a Portal, go with a Customer Portal. You will need self-registration functionality which I don't think is supported by Partner Portal.

How long is salesforce going to support said portal? (I saw some talk of self-service being outdated)

There are no plans to cease support of Salesforce portals. Existing portal customers will have Communities functionality provisioned at no additional cost

Once I get the portal created what's the best way to link it to salesforce.com?

Playing around with the Login Settings on a custom subdomain in a Developer Edition org. Try out the High Volume Customer Portal profile and create Sharing Sets to expose your Salesforce data to visitors based on linkages of records to their associated user's Contact or Account.

Note that the going rate for Customer Portal (Community) licenses are about US$5.00 per user per month. The going rate for Partner Portal (Community) licenses are about US$35.00 per user per month.

You may be expected to purchase a minimum headcount (perhaps 100 for Customer Portal, 20 for Partner Portal) at a minimum subscription period (perhaps 12 months).