Welcome to the forum and to the wonderful world of Betta keeping....
LTB has covered some, however, let me expand on that a bit....
The nitrogen cycle will happen without you doing anything as long as the needs are met for the beneficial bacteria colony-Here is a sticky to help you understand it more-ask question on what you don't understand
http://www.bettafish.com/showthread.php?t=107771
You do need a thermometer to monitor both the tank temp and temp of the replacement water used for water changes.
You may or may not need a heater-Your goal is to maintain a somewhat stable temp in the 76-80F range.
Since your tank has a filter-you may need to baffle it, however, if he is inactive and/or hiding right now you might want to turn if off for now and once he adjust-then re-start it later in the week.
What kind of live plant did you get-can you post a pic of it....
How did you acclimate him to the new tank, what is the current water temp, type of dechlorinator and source water info...well water, tap water...etc.....and is the source water used-does it go through a water softening unit on the house that uses salt, if so, can you bypass it for the water used for the Betta.
Bettas generally will adapt to your source water pH as long as extremes are avoided-Its best not to try and change the pH using any chemicals-usually you end up with pH swings and this can be more stressful.
Nutrition-good quality varied diet fed in small frequent meals-Its not uncommon for them to not eat for a few days to weeks when you first get them-Keep offering food and remove any uneaten food within a reasonable time.
Lovely fella by the way....