Raise, lift, as in We have to take up the old carpet and sand the floor. [c. 1300]
Reduce in size, shorten, tighten, as in I have to take up the hem of this coat, or You have to take up the slack in that reel or you'll never land a fish. [c. 1800]
Station oneself, settle in, as in We took up our positions at the front. [Mid-1500s]
Accept an option, bet, or challenge, as in No one wanted to take up that bet. This usage is often expanded to take someone up on, as in You're offering to clean the barn? I'll take you up on that. Take up dates from about 1700, the variant from the early 1900s.
Develop an interest in, begin an activity, as in Jim took up gardening. [Mid-1400s] Also see go into, def. 3.
Use up or occupy entirely, as in The extra duties took up most of my time, or This desk takes up too much space in the office, or How much room will your car take up? [c. 1600]
Begin again, resume, as in I'll take up the story where you left off. [Mid-1600s]
Deal with, as in Let's take up these questions one at a time. [c. 1500]
Absorb, as in These large trees are taking up all the water in the soil. [Late 1600s]
Support, adopt as a protegé, as in She's always taking up one or another young singer. [Late 1300s] Also see the subsequent entries beginning with take up.