Sure...consider a few different styles of camera.
If you want to stick to fixed lens cameras like your Dimage, the SX20 is worth looking at for sure...though there are others worth considering which can perform as well, and possibly better in a few situations.
If you want to consider slimmer, more pocketable cameras - add the Panasonic TZ10 to your list, and the Sony HX5. Both are worth considering, for having flexible zoom lens ranges in compact bodies. They can go 10x or so, which is pretty decent for most uses, but shorter than the 'ultrazoom' cameras. The Panasonic out of the 3 probably has the best lens built onto it...the Sony probably has an edge in burst-speed shooting and in low light ability. Both are fine cameras.
If you'd be willing to deal with something a little bulkier like the SX20, consider some additional ultrazoom type cameras. Ones worth considering might be the Panasonic FZ35, Fuji HS10 or S1800, and maybe the Pentax X90.
Again, the Panasonic lens is probably the overall best here, while the Fuji HS10 is interesting for its traditional-style zoom/focus ring on the lens barrel, like the big-boy cameras. All are fine cameras too - not much to complain about between any of them.
DSLRs would be a much bigger step-up and commitment for you - something to consider if you think you're going to get pretty serious with photography - they will all perform faster, focus faster, shoot better in low light, and shoot action with better faster focus and shot-to-shot times. But they are significantly larger, heavier, and require additional lens investments for various purposes. Many will recommend DSLRs to you, but you need to decide if you really think you need one, or want the extra weight, size, and expense.
Mirrorless interchangeable Lens cameras are an interesting consideration if you want better than P&S quality and speed, but don't want the big, heavy DSLRs. They still require a lens investment, but are smaller and lighter than DSLRs, and function a bit more like P&S cameras using LCDs or electronic viewfinders to shoot, rather than optical. Ones to consider would be the Panasonic GF, Olympus EP, and Sony NEX series cams.
