Lasso is one of 9 different ways I can think of to select things in GIMP. If its getting pretty close, and it looks like it is, the next step would be to edit the selection before you take a copy. Or mask out the edges of the selection you don't need after you have placed the selection.
The method is similar for both, but you can undo mistakes easier with the 2nd one.
If you want to edit a selection manually, first select something. On the bottom left of the edit window is a small square button, press this, or the key combo Shift+Q. What you have selected will be Clear, what you have not selected will be Red. Now if you select the paint brush (p) you can paint things that you want selected. Black paint adds and white paint removes from the selection. You can zoom right in and get as precise as you like. Interestingly filters like Gaussian blur work in this "Quick Mask" mode, so if you want to blur the edge of the selection apply a Gaussian blur. It won't blur the image just the selection paint. When your happy with your fine tuning of the selection, press that small button again, or shift+q and it will go back to normal mode. Then copy and paste as before.
Make your rough selection keeping a bit more than you need. When you paste this as a new layer, right click the "add layer mask" and choose "white full opacity". You will see two layers side by side, one is your image the other is a full white block.
Now anything you paint black will just put a hole through to the background and become transparent, anything white will stay in front. So paint around the rough edges with black paint. The rough edges will still be there, but they will be invisible so you will see the background. If you go to far no worries, paint some white paint on the thing you want to be visible again.
Pretty clever stuff. You can switch between editing the Mask and editing the actual layer by right clicking the layer and toggling "edit layer mask". If you really want to have fun you can play with greys for semi-transparent images.
This is how people do selective colour photos, they have the background as a colour image, they copy that to the foreground and convert to black and white, then they apply a mask over whatever they want in colour so that you see through the black and white to the colour layer for that item.
Good luck. If you get this you basically have learned a good deal about layers and masks in a very short time.