It depends. If you do not have a google account, then google will match your searches etc to your ip address. If you move to a different address, then that data collection process gets started again.
However, if you have a google account, then your search history and other statistics get linked to that account, and will thus travel with you to any other computer if you log in to that account.
Also, google in general doesn't "show your data around", it keeps it for itself. However, given that google controls a vast share of the internet advertisement market, they will use that data to tailor the ads you get to see to your profile as determined by your recent searches and the sites you have visited. For example, I did quite a bit of research into smartphones prior to me acquiring one, as a result, I get a higher than average number of phone-themed adverts at the moment.
And yeah, if you start moving to other search engines like Microsofts Bing, pretty much the same will happen; it's just that google is such a large player in the web that it's more readily apparent.