To collect blood you need:
-Research journal in your inventory (craft it from empty journal or notebook and pen or pencil, DONT read it)
-Scalpel
-Test tubes (find them in the world)

Right-click on dead body in the world (not in your inventory) and select the option to get a blood sample.

To develop an antiserum/cure you need some items:
1) Injector - find them in medical containers or anywhere in the world
2) Chemicals - find them in medical containers or anywhere in the world
3) Mobile laboratory - rare loot to find or you can also craft it yourself (Electric skill: 2).
4) Blood samples (fresh) - collect them from dead bodies. One body = one sample.

Blood samples last for only one day before going stale. However this can be extended significantly with a refrigerator and freezer. I highly recommend freezing you samples until you need them. In game terms, the samples are just like food items, so how long they last refrigerated depends on the refrigerator effectiveness settings of the server. I play with high efficiency fridges, so feedback in this area is appreciated if everyone feels it is still too fast of a decay.

Check for Infection:
You can check if you are infected anytime. Just collect those:
-First aid skill: 2
-Empty Injector
-Chemicals
-Chemical Workstation

Weak Antiserum:
Works if your infection level is between 20% and 70%. Reduces your infection progress to 20%. To craft it you need:
-First aid skill: 3
-Blood samples: 10
-Chemicals
-Chemical Workstation

Strong Antiserum:
Works if your infection level is between 1% and 85%. Reduces your infection progress to 1%.
To craft you need:
-First aid skill: 5
-Blood samples: 20
-Chemicals
-Chemical Workstation

The Cure:
Works as long as your infection level is below 95%. Cures you from zombification.
To craft you need:
-First aid skill: 7
-Blood samples: 50
-Chemicals
-Chemical Workstation