There is no real way to avoid it, you will end up taking a lot of damage in Grounded. Bombardier will spit goo at you, Spiders will stomp on you, and swarms of ants will chase you around, nibbling at your heels. It all adds up, and in your early hours, you will visit the respawn screen a couple of times at least.
To avoid death, you will want to get your hands on some bandages, and the more efficient upgraded bandages that cost fewer resources to make. To do this, you will need to get your hands on the Anthill BURG.L Tech Chip. Because you will be heading for an area with lots of ants, it is also a good idea to get the Ant Armor Set, as this will allow you to get near to ants without them attacking you.
To find the chip, you will want to go down into the Anthill, then take the first left, which will have a number of roots hanging out of the ceiling just in front of it. From there, keep traveling down the long tunnel and then you will come to an intersection of tunnels. Take the second tunnel on the left, which will lead to the room that contains the chip. You can find it lying against the wall, straight across from the entrance to the room.
After that, you need to return to BURG.L in the Oak Tree Lab and give him the chip. His shop will now have an item called Fiber Bandage Efficiency that you can purchase for 3000 Raw Science. This will reduce the resources you need to make bandages to a single plant fiber, and one sap.