I have got a tdnam membership. I saw a domain name yesterday and its price was $10. So if I bid $10 or more then $10 then will I get that domain name? And if I get that domain name will it go to my Godaddy account or any other registrar? Are there any other fees involved in this which I will have to pay to tdnam or the seller?

If it is an auction, which starts at $10 and you bid $10, and no-one bids higher than you, then yes you will get it.

Domains at other registrars can be listed at tdnam, but I have never come across any - i have bought and/or sold around 100 domains on tdnam.
it's always best to check the whois to see where it is registered.

If the domain is an expired auction, then yes, godaddy will put the domain straight into your account after the redemption grace period has finished.
(assuming the old owner does not renew)
You will have to pay a 1 year renewal.

If the domain is being sold by the current owner, then you will contact them after the auction ( or they will contact you ) in order to arrange the account change / or push / or transfer.
In this case you will not have to pay extra fees, unless you decide to transfer the domain from one registrar to another. The usual process is just to push the domain to your account at the same registrar.

I also have membership but I usually go there to sell my domain names I purchase the expiring domain names from pool.com and namewinner.com