The beneficial bacteria that make up the nitrogen cycle of a tank - the ones that process the ammonia into nitrite/nitrate do not normally live in the water column. There are times when you will find substantial numbers in the water - as in a "bacterial bloom", but this is rare. In general, the BB fasten themselves to all the surfaces of the tank where they can find access to the waste (ammonia/nitrite) and oxygen - the things they need to do their work.
If you provide sufficiant "inviting" platform for the BB in your filter box, 98% of the bacteria colony will be found there. There will always be a smaller number of them found on the deco, tank sides, and gravel base, but most will be where the majority of the waste and oxygen are - the filter. This is why it really pays to carefully plan and load your filter well - and keep it clean and running efficiantly.
The gravel at the bottom of the tank will hold some BB. But the BB can only colonate on the surface of the gravel - underneath does not get adequate circulation of the water - carrying waste and oxygen. So taking a large scoop of gravel to start a cycle will only work so-so. Carefully removing the top 1/2 inch or so in a fair area will do a better job. Using water from a cycled tank will not do much of anything to start a new cycle in a new tank. It might speed things up by a day or so - but not enough to make it noticeable or worth it. Far better to simply start the tank out "clean".
I love bare bottoms. Gravel bases take up valuable water volume - water the fish could be using. They also trap waste. No matter how carefully you vacumn, there is ALWAYS waste left in the gravel. Gravel is also a hiding place for parasites in various life stages.
A bare bottom tank does not need gravel to hold the BB. Use a filter that gives at least 10 times turnover of the tank per hour (a 36 should have AT LEAST 360gph turnover - I put at least 400gph on my 30 gallon tanks) and fill the filter with plenty of platform (media) for the BB to colonate. This can be sponges, ceramic cylindars, sintered glass, bioballs, or floss. A biowheel is a nice addition, too, but I do not ever depend totally on one. It is easy to fill a small net bag or tied stocking full of sintered glass or ceramic noodles and drop it in behind the cartridge or under a sponge or such in a filter to increase the platform available to the BB.
If you wish to jump start the cycle on a new tank, the easiest and most efficiant way to do so is to purchase the FILTER for the new tank first. Set the new filter up on your old tank and set it running. I take the media from the old filter out of it and split it in two piles. Each pile is mixed with enough "new" media to make up the volume of media needed for each filter. Each filter - new and old - are then stuffed with the mixture of seasoned media and new media and set running on the old tank. With media containing BB in contact with new media, the bacteria transfer easily and readily. Running the filter on your old tank for a couple of weeks should set the filter up nicely.
When you move the new filter over to the new tank, you will experience a small bump in cycle in both tanks. Your fish only provid enough waste to support a certain size colony of BB...... when you take 1/2 of it away, there is "less than" what is needed to process all the waste. BUT - since the colony is all there - just not quite big enough, within a day - or a few - your BB colony will grow to handle anything it finds - and your cycle can be complete within a few days.
Better yet - if you are abandoning the old tank - move BOTH filters over. The tank will be cycled ... and when you add a new fish... it will grow easily.
If your original tank is simply not capable of supporting another filter, you can take the old filter off and run just the new filter on the old tank - stuffing it with all the media. IF you cannot do this, you can still stuff a bag or stocking full of media and put it in the old filter - or suspended under the outflow of the filter. If this does not work, place the bag of media over an airstone so that the air stone bubbles up through it. This will pull water and oxygen through the bag of media - even though it is on the floor of the tank - and cycle the media fairly well.