On guitars that I play regularly, strings get changed every 4-6 weeks. On guitars that I play infrequently, it's probably closer to every 4 months.
As per Elixir... It's true that they stay new sounding MUCH longer. Especially for those of us who tarnish metal quickly. I used to swear by Elixir when I wanted new sounding strings all the time, BUT didn't want to change them more than a couple times a year. Now I enjoy the process of changing strings, cleaning the fretboard, and polishing the frets.
Nothing kills tone more than dead strings. If you're the type who doesn't want to change strings more than 2 or 3 times per year, try Elixir. If you don't mind changing strings more often, explore your options. Personally, I like D'Addario Phosphor Bronze for my acoustics, and D'Addario Nickel Wound for most of my electrics. They sound great and stay new sounding substantially longer than Ernie Ball/GHS/Dean Markley/Rotosound. While I much prefer the tone of my D'Addario strings (during their optimal service period) over Elixirs, if I was going to play them long past their prime, I'd probably go with Elixir. I'd rather have an old set of Elixirs over and old set of anything else. But if we're talking reasonably new vs reasonably new; there are countless strings that I'd take over Elixirs.
Edit: I should add that another important reason to change strings frequently is tuning stability and intonation. Old strings do not stay in tune, warble all over the place, and do not intonate well up and down the neck.