How can I transfer my Terraria house to a new world?

I recommend using Tedit. It has a tool that allows you to copy part of a world and save it. The exact steps to using said tool are as follows:

1. Download and install a suitable version of Tedit. If working with a 1.3 world, I recommend using the experimental version here. It seems to work okay without corrupting worlds, but nevertheless you should backup world files before editing them
2. After backing up the world you are copying from, open it in Tedit and switch to the selection tool (in other words, click the blue square on the left side)
3. Select your house (or anything else you may want to transfer) just as you would if you were selecting files with a drag-box.
4. Click the copy button (the two overlapping sheets of paper) on the left toolbar to copy the selection to Tedit's clipboard.
5. (Optional) delete the selection area so you're not tempted to open the old world and double your items. If you do this, make sure to save, naturally.
6. Switch to the "clipboard" tab visible on the right side of the application
7. At the bottom, there should be an option to export the contents of the clipboard. Click it and save the file somewhere you will remember
8. After backing up the world you are copying to, open it in Tedit and go to the clipboard. There should be an option to "import schematic or image". Use it to import the schematic you saved in steps 4-7.
9. Click the paste button (clipboard with a sheet of paper on it) on the left side and drag to where you want the house (or whatever it is). Be sure you get it exactly where you want it.
10. Save the world! (Sorry, I had to make that joke.) Don't close Tedit, in case you want to undo the paste and adjust the placement of your house/whatever
11. Open world in Terraria and see how you did with the previous steps. If necessary, go back to the editor, undo the pasting of the house and place it again.
12. Once satisfied, congratulations! If you want, you can repeat steps 2-11 (aside from backing up. A second backup would probably be a waste of time, though it's ultimately your call I suppose) to copy more sections of the map. Just don't go overboard (or do. It doesn't matter to me)

updates: this solution should still work for Journey's End (Terraria 1.4), but you'll obviously need a compatible version of TEdit. Google is your friend, as is this page