Follow and Chase

Follow

To follow means to go or come after something or someone, be it in terms of time, sequence, or direction. To follow can also mean obey, or in certain situations to accept someone or something as a leader or guide.
Ex: As Kevin did not know how to get to Erin and Christian's house, he followed them in their car the whole way there.
Ex: The chemistry students followed their teacher's instructions very carefully so that no experiment would result in anyone getting hurt.
Follow through means to carry an act out to its completion. To follow up means to make something more effective by pursuing further action in the future, for example, a salesperson trying to make a sale would follow up on initial conversations by calling potential clients again.
Ex: Chelsey made a lot of plans to save money to visit her friend Paula in London, but she never followed through with them so she never got the money together.
Ex: An important part of making sure that marginalised populations receive the social services that they need is not just visiting their communities, but also following up after the visits.

Chase

To chase means to pursue or to run after someone. In a more figurative sense, to chase means to be persistent in asking someone for something.
Ex: When he saw the police arrive, the suspect immediately ran. The police chased after him and in the end caught him in an alley nearby.
Ex: Steve works for a debt collection agency. His job is basically to be on the phone all day chasing after people who have not paid their bills.
To give chase is to head out in pursuit of someone or something. A wild-goose chase is an expression that means that something is a pointless or worthless chase.
Ex: The hounds quickly caught scent of the fox, and they gave chase. The hunting party followed soon after.
Ex: My boss sent me out on a wild-goose chase for the hard copy of that contract. I looked all over the office before he realized that it was on his desk the whole time.