As you've only just started cycling, now is the time to change over. You won't have many bacteria yet and although it may cause a hiccup in the cycle, that's all it will be. Use the sponge from your current filter. Cut it up if necessary to make it fit the new filter then cut up the media (sponge, ceramic noodles etc) to fill in any gaps left. Once the tank has been running several months you can then remove the old sponge a bit at a time and replace with media from the new filter. I would wait at least 6 months before doing this to allow the media to mature, then swap a bit every month till it's all changed to the media that comes with the new filter.
If you are doing a fishless cycle and have the patience, you might want to consider just starting again from the beginning with the new filter. But if you are doing a fish-in cycle, you will be better preserving as many bacteria as possible.
The bacteria in the filter are lot hardier than used to be thought. With an internal filter, you need to turn it off when doing a water change because when the water level drops enough that the impeller will be above the water, it damages a motor to run it. Unless you take hours and hours to do a water change, just leave the filter attached to the glass till you fill the water back up again. The media won't dry in an hour or so, it is quite safe leaving it that long.
And on the subject of turning things off during a water change, you must also turn the heater off as they can shatter if they switch on out of water. You just need to remember to turn them back on after.
Cleaning the filter, I just take the filter out of the tank and put it in a bucket, then siphon water out of the tank into the bucket, take the sponge out and squeeze it to get the muck off, then put the filter back in the tank and carry on with the water change. The first few times you do it, you'll find a lot of mucky water drains out of the filter back into the tank while you are lifting it out. Don't worry about that, just siphon the muck out. You'll get quicker with practice
Plain sponges do not need to be replaced like the manufacturer will say. Sponges only need to be replaced when they go holey or won't go back to shape after washing. And when this happens in a few years' time, cut the old sponge up into at least three, then replace one part every month till you have all new.