Does an investigator have to perform both actions before another investigator can go, or can they split them. For example, X has a healing ability, but is too far away from Y. Can X move towards Y, then Y moves towards X, then X uses the ability now that Y is in range?
Can investigators split their actions?
Nope. Investigators take discreet turns, using as many of their actions as they can/wish, then their turn ends and they do not get another turn until the next round.
If Carson Sinclair is present, however, his special action could allow another investigator to take an action 'outside' of their turn.
Carson is nice with 4+ Investigator...he can save the day alone. I really like him with 4+ Inv but under he's not that useful (uniquely from my experience, ymmv).