You can move everything straight over from your 30l once it finishes cycling this includes decor gravel and FILTER. Then you run your normal internal filter along with your new filter for around 3 months and you DO NOT add any new fish during this time. After 3months you can remove your old filter if you so wish to.
If I remember your current tank is quite poorly stocked? Don't you have a common Pleco currently in your 30l?
Unfortunately a 125l tank will still not be big enough for this fish.
But the move can happen pretty seamlessly.
I would definitely make sure you are able to test your water parameters using your own testing kit to make sure your levels remain ok when you move everything.