What the best way to put splitters. Right now I
have 2-way splitter at the entrance. One end goes into family room
with TV and another goes to office with cable modem. I need to run
cable to three more bedrooms upstairs. The best would be to replace 2-
way splitter with three-way splitter and run one end to the artic and
then put there three-way splitter and run cable to each bedroom. Will
it work?

It will work, just remember, a 3-way splitter has two -7db outputs
and one -3.5db outputs. put the cable modem on the -3.5db output as
you will want the least signal loss for your cable modem and also if
you have VOIP phone service. The TV signal is not as critical

Also what kind if splitters are the best?

As long as they go up to 900 MHZ or better, you should be fine.