Well, you first need to get HM1 (Cut) and teach it to one of your Pokemon. You also have to beat the second Gym before a Pokemon will be able to use it outside of battle. Then you just go up to the tree and press A, the Pokemon you learned Cut will cut the tree down.
PS. If you are having trouble finding HM1 you should check out the SS. ANNE in the port in Vermillion.
You have to get the HM Cut from the Captain of the ship in Vermillion City.
You get HM01 Cut from Ilex Forest. Use that to cut down trees.
bidoof and bibarel can cut trees
with TM cut
Teach HM Cut to a Pokemon and use it on the tree.
HM01 Cut
You have to get the hm, then, you have to beat bugsy, the Azalea town gym leader. After that, teach it to one of your Pokemon so you can cut down trees.
If there is a pokemon in your party that can use Cut and you have the second badge then face a tree that can be Cut down press A on it then press A on YES to cut the tree alternatively you can open the pokemon screen and select your pokemon and press A on Cut.
By teaching a Pokémon HM Cut, and then using it.
Heracross,Burmy(leaf,trash,and rock),Wurmple,Silcoon,Munchlax.
Teach it cut, you get it on the side of viridian forest, below diglett cave
Use 'CUT' on the door.