You can look at the whois information, and get an email or phone number, then contact the owner. The only problem with this approach is that since you are contacting them, they will raise the price (big time).
Your best bet is to find if the domain is for sale by googling, or go to the site and see if there is a link that says it is for sale.
I would recommend getting a domain name that is available, then moving over to the domain name that is already taken (e.g. DropBox was getdropbox.com, then moved to dropbox.com).
EDIT: Check out this link.
I would recommend sticking with domains that are instantly available - either free to register, or can be purchased with a credit card on the spot.
I've tried doing a 'whois' and doing detective work to track down domain owners probably between 10 and 20 times. The majority of the cases, I got no response. Of the ones that responded I made an offer and got no subsequent response. The one time I actually settled on a price, the transaction fell through because the owner lost the password to their registrar account.
Maybe I've just had bad luck, but I suggest not wasting your time trying to get a domain that isn't directly for sale.
Instead, I would spend your time coming up with interesting alternatives.
I use instantdomainsearch.com (as-you-type availability checking, plus some links to high-end domains for sale).
Or if you have money, go to a domain auction house like sedo.com and see what they have for sale.
GoDaddy has a 'bulk domain search' where you can type in up to 500 domains and it will tell you which ones are available. I reccommend buying a 'new' domain, not one that is already owned by someone else.
The biggest hurdle to buying domains on the secondary market is not understanding pricing and value in that market. If you really like a particular name, the first thing to do is determine approximately what the domain is worth.
If there's no price on the domain itself, first check one of the reseller sites, like SEDO or Afternic to see if it's listed for sale. These sites also have people who will help you negotiate a sale (for a fee.)
If it's not listed with a price anywhere, check someplace like namebio to see if there's a sales history of similar names. If you're lucky, you'll find history for the domain you want. Another thing to try: estibot (an automated appraisal tool - can be wildly inaccurate, but for lack of anything better can provide you with a starting point)
Once you get some ballpark idea of value, assuming it's within your budget, figure out the max you're willing to pay and make an offer (leave some room for negotiation.) You can get the registrant's contact info from any whois lookup. Assuming the name is for sale, reasonable offers usually get a reply. Lowball offers will be ignored. If you don't hear back or they come back with a price that's out of your budget (there are some idiots who will jack up a price as soon as they get a nibble), move on.
Of course registering a fresh domain is the easiest and cheapest route. I would avoid unusual extensions - if you haven't heard of it neither have your visitors. Sometimes weird esxtensions get lower clickthrough because of a lack of trust. Stick to .com/.net/.org or your country code (if you're outside of the US)