By being prepared.

Swamps is for most players the hardest step to take. You need to be careful. You need to have the best food you can get. You need to have poison resistance potions!

Draugrs hit hard. Very hard. Have a safe base at the bordering biome. Start scouting from there at the borders. Always have a poison resistance potion in you. Leeches and green blobs can and will poison you. If you're getting poisoned and have numerous draugrs on your tail, you're going to be dead meat.

Have a hoe with you and make paths, so you can avoid leeches and don't have to swim. You will need the bloodbags of the leeches later, as well as the entrails of the draugr, so better keep them.

Beware of the frumious bandersnatch... err, abomination, I mean.

The swamp can be tamed with the right armor, shield and a mace. Not the abomination. You need an axe for that. But you want it. The first thing you should create from the roots of an abomination is the mask.

It will give you poison resistance. Whenever I enter the swamps, the mask will be my faithful companion. You don't need the iron cap. The root mask is your friend.

Edit:
The first abominations need to be killed with fire arrows or by moving it over the surtling fire geysirs. The first item you make from iron is the iron banded shield. It will block most of the abominations attacks.

Edit2: I still find the swamps the most horrible biome to venture into. Everything else seems easy against it. Even the dreaded 'squito is a joke against a two star draugr archer when you're still in troll armor.