Transfering water from your old tank to the new will do nothing to help the whole bio thing. If enough is transferred it may help a little to acclimate the fish from your old tanks water parameters to the new tanks water parameters, but wont help your cycle at all. Beneficial bacteria lives on surface area, not the water column, so transferring water wont transfer anything of benefit at all. Would be better off transfering filter media (or the whole filter) from old tank to new.
As to algae, your new tank will be cycling all over again so expect some. Brown algae (diatoms) are normal in a newly cycling tank and normally clear up in time without intervention. Increasing light can help with brown algae as it promotes green algae which outcompetes brown and is easier to deal with.
Green algae is caused by an inbalance in nutrients, light and available CO2. The best way to control green algae is finding a balance between enough light for your plants but restricting it enough to reduce algae growth to a managable level. 6 to 8 hours a day of lights is normally the sweet spot people arrive at. More plants will take up available nutrients and reduce algae growth also. Regular water changes as well.
