All of the services linked here can be transferred with you if you move between institutions.
All you will need to do is update your details (current post/institution/etc.) and in particular email addresses if appropriate.
Web of Science have a service called ResearcherID. You can create an account, which comes with a a (free) unique personal identifier. You can add your papers to your account, track citations of those papers in the Web of Science. The profile can be public or private and also tracks metrics about you like the h-index.
Using ReseacherID can make finding your work easier in an authors search. I.e. filtering your work from those by people with the same family name and initials. This also can make finding bibliometric impact metrics easier, and more accurate.
ORCID allows you to register for a persistent ID for yourself.
Google Scholar has a similar service called 'My Citations' service - you can create a profile and track your publications' citations.
Unlike ResearcherID, the Google Scholar service can attempt to update your profile automatically. It finds citations that it thinks belong to you, and can add them automatically to your profile. Alternatively you can request it sends you an email to confirm you want to add the item(s).
This is an emerging movement to use a wider range of metrics in addition to traditional citations. This can include discussion (e.g. Twitter), saving (e.g. Mendeley, Delicious) and sometimes viewing or recommending an item.
This can give a better picture of impact, but until this is better understood probably needs to be used with caution.
We have a lot more information on our guide to altmetrics.