Good morning BinSkiBoi. I agree with most of the above statements. Moving your filter from the old tank to the new tank will give you the instant cycle. Once you move the filter move the fish and all of the ornaments you want to move. It would help too, if you added some of the gravel from the old tank to the new tank. You can place some in a "never seen soap" nylon hose and place it in your tank for the few weeks as well. (at least 2 weeks and as stated 4 is better

) Keep all of your filter media wet and your gravel and ornaments wet too.
As for transferring your water. Michaelb is partially correct. The good bacteria in your tank needed for the cycle is not free swimming but attached to all surface areas of your tank. That is why it's suggested that you "not" transfer the water. However, I suggest transferring enough water so that your fish can go into the new tank without having to
acclimate them all over again. If your water chemistry,
ammonia, nitrites and nitrates are where they should be 0 0 5-20 in the 10g tank then I would use some of it as I mentioned above. (If not then I suggest acclimating your fish to the new water.) Add the old water first if your readings are good, then the fish, then the new water. Dechlorinate the new water first before adding it to your tank. Check your
pH levels and temperatures and try to make sure they are as close together as possible. (compare the old tank pH and temp to the new water ph and temp).
Congrats on the upgrade BinSkiBoi. Welcome to Fish Lore! Keep us posted on your progress and I hope you can share some photos with us along the way.
Ken