'Take up' can mean to begin or start something. This usually refers to something which is ongoing, such as a hobby, job or role. You can also 'take up' time; space; resources; or attention, meaning to occupy or use them up. If you 'take up a topic' or 'take up a conversation', it means you resume from where you left off or return to something you were previously discussing. An older use of the phrase 'take up' refers to sewing up the hem of clothing, such as a skirt or trousers, in order to make them shorter and a better fit. Note that you also turn this phrasal verb into a noun by inserting a hyphen: "There hasn't been much take-up on the redundancy offers. We'll have to convince people to take us up on them."
Additional examples with 'take up' are:
An elderly man is discussing his plans for when he retires:
"When I retire, I'm going to take up gardening to keep myself busy."
Research into productivity in the company has revealed some surprising results:
"Our research shows that over 40% of our employees' days are taken up by meetings."
Your friend is complaining that her partner doesn't spend enough time at home with her and their children:
"I don't really know what advice to offer you. You should really take this up with your partner."