On your first night, you are at your weakest. In the best-case scenario, you have already collected a decent amount of wood and stone, some type of food source, and you have killed several sheep for a few wool.
Build yourself a home. Typically, a first home is build out of a combination of wood and stone, depending on how you spent your first day. The easiest way to create a safe home is to mine into a mountain and create a door to the home. (On hard mode, just place a block down instead of a door to avoid zombies destroying the door)
In the end: Use a bed. If you have the 3 wool to spare, a bed can save you a great deal of time by allowing you to bypass the night (Requires a small lit space, however).
Stay on the move. The spawning algorithm of minecraft typically leaves monsters a fair distance away from you, at least at the beginning of night. As a result, you can typically avoid monsters in forests while cutting down trees. Note: Keep an eye out for any potential enemies. Creepers can sneak up behind and you will deal a large portion of your health. You can also play aggressive and kill monsters in the vicinity before you work on a tree. I typically look for a suitable biome to build a base in at night, using sprint to avoid monsters (Double-tap W)
Rather than attempting to cut down trees outside, begin work on a mine that descends to an elevation of around 12 (One of the most productive diamond levels). Wood, while useful, is typically only used for chests/sticks in terms of mining, and sees very little use. This will allow you to work on Iron Armor/Iron Swords while waiting for day time.