Backpack
Crafting a backpack as soon as you are able to will help you immensely as you do not get a lot of inventory space to carry food and medical supplies. The leather you need for crafting a backpack can easily be found by killing wolves. Also, having inventory space open will enable you to bring back whatever you might find.
Furnace
Make sure you have a furnace as it can be used to cook meat and also process raw ore for use in more advanced tools. It is important that you are able to cook raw meat because one piece can almost fill your hunger meter entirely. Before you head out exploring make sure you load up on cooked meat.
Medical Supplies
It is important that you carry medical supplies at all times. Plenty of healing items can be found on bushes and will be needed as some status effects will not go away until they have been properly treated. Regular damage you will heal from naturally as long as you are not starving. Bandages will be needed to deal with bleeding and radiation pills are necessary for when you wander into a dangerous area. You may even want to consider taking resources that will enable you to craft a splint so you are able to stabilise any injuries you may get while the body heals.
Weapons
One thing you will need to take with you to have any chance of surviving is a weapon of some kind to defend yourself against enemies and predators. Look to either find or craft a ranged weapon as they allow you to take out the danger before it can reach you and avoid bleeding wounds.
Maps
Always use the maps in your journal to guide you to locations and objectives. One of the maps you have access to provides a view of the entire island while the other gives you a local snapshot of the area. To zoom in on the island map you just need to tap on it. You can waypoint on either map by tapping wherever it is you would like explore.
Base
You may want to consider finding or building a home base which you can return to when it gets dark so you can store all the things you collected while you were exploring. If you decide to build a base it will take a large amount of planning and materials but more importantly regardless of whether you decide to build one from scratch or refurbish one of the existing 'abandoned' buildings that can be found throughout the island you will need to weigh up the pro's and con's of the location before you place anything.
Torches
One essential piece of equipment you need when you go exploring is torches as you do not want to be away from camp in the dark without a light source. Torches can either be found or you can craft one yourself by using wood, stone, and fat. You will be able to find fat on sheep and bears. Make sure you have enough wood in your inventory so you are able to build a camp fire at night as this will ward off some animals and also keep you from freezing.