I have to agree with the other answers as regards DES - it provides little defense against a motivated attacker.
In terms of RSA (the other algorithm you mention in your title), public-key encryption algorithms are generally considerably slower (by a magnitude of about 1000 based on what I've read, although I've never personally timed it).
It's also arguably less secure to use public-key cryptography for exchanging long messages.
As some background, public-key cryptography generally depends on some kind of a trap-door function (i.e. a function that is relatively easy to compute but is difficult to find the inverse for). It turns out that those functions are remarkably difficult to find; one of the most common ones now (which is what RSA is based on) is integer factorization, which is NP-Intermediate on "standard" computers (but is broken for quantum computers).
First, the fact that integer factorization is NP-intermediate is at least a theoretical weakness in RSA - technically, no one's actually proven that NP-intermediate problems are intrinsically more "difficult" than polynomial-time algorithms (although it's widely believed that they are) because that would entail solving the P vs. NP problem, which is one of the major outstanding questions in computer science.
It turns out that many of the trap-door functions aren't quite as difficult to find inverses for as it is to break a good symmetric-key algorithm like AES or Twofish - i.e. the best public cryptanalysis for public-key encryption algorithms tend to be at least somewhat more feasible than the ones for symmetric-key algorithms. (There's a good article here on why it's completely infeasible to "break" AES with brute force, and the known attacks against it aren't even close to feasible either).
For this reason, public-key cryptography's often used for things like key exchange, at which point both parties switch to symmetric-key cryptography.
All that to say that the other people are right - use AES :).