I use Trillian and eBuddy. eBuddy is more stable but Trillian looks like it will be better once it gets out of Beta. eBuddy you can get from the Market but here is the link for Trillian Trillian for Android.
I have heard of one other called Beejive. I think it's in Beta as well but never tried it.
Oh I didn't read it that way. LOL I thought you were talking about IMs but now that I read it, it does sound like you are talking about SMS. If you are talking about SMS, when you click on the picture the first time, it enters slide show mode. When it is in slide show, click on it again and the bottom right button will save it. If that is not there, then download Save MMS from the Android Market.
If we're talking IM, I use eBuddy. I used it on my iPod touch (when I had it, and was into that kind of thing), and so far so good on the Droid. I used another app before that was 293847239847239847 times better than eBuddy, but I forgot the name
If we're talking SMS, I found one last night that I really, really like... chompSMS. It's like bubble chat, you can customize colors, notifications, and LED light... pretty snazzy.