Which camera lens would be best for me?

I'm receiving a second camera lens as a gift from a family member and I don't know which one to choose. I have a Nikon D3100 and the standard 18-55mm lens at the moment. I would prefer a zoom lens, but then the other thing to consider is I'm taking a trip to China with my school in about 6 months and I don't know what lens I would take. I couldn't spend all my time swapping between lenses, but a 18-105mm or a 16-85mm lens is unfortunately out of my price range. Since its a gift from someone I don't know too well I don't want them to have to spend outrageous amounts. The one I was originally thinking of is the 55-200mm lens, but then there's the whole changing lens thing.
I was wondering if anyone had any suggestions for what lens to go for? It needs to be $250AUD or less preferably, a tiny bit over could be fine. Are there any screw on zoom attachments to my original lens that could do the job?

You either have to do the whole "changing lens thing" like all of us pros do, or you can buy the Nikkor 18-300 mm lens, but that will cost you nearly $1, 000 USD

For great quality I had a 18-200mm and found it too heavy to lug around. I got the 50-200mm which is almost exactly the same physical size as my 18-55. The only way I could get lighter is give up on the DSLR and buy a bridge camera! This is the perfect setup for me!

Go for the 55-200mm. While you will have to change lenses, it probably won't be that big a deal. Most of the time you either favor one lens over the other, or will be shooting photos at any one given time using one or the other.

In other words, lens changes are usually minimal.

For instance, say you are in an urban environment. You will probably put your 18-55mm lens on and not need the 55-200 at all. Then say later that day, you go to a zoo and want to get some closeups of the animals. You would probably change out to the 55-200mm and use it the entire time you are at the zoo.

Most seasoned photographers are not constantly changing their lenses, but only to meet the needs of the situation at hand. And as long as they are in the same situation, they don't usually change lenses.

Nikkor 55-300 is a a great lens for around 300-500 USD but really can't get anything that is really good for less than that. I have the Nikon D3200 and I use it. It focuses really fast and you can get some really awesome shots from really far away.