As someone who has little life and a large penchant for daydreaming, I've come up with a rough list of things, in order of priority:
1) Hire an accountant. I would do this before I even collected my new-found wealth
2) Pay off all of my debt
3) Gift large sums to my family, working with my accountant to see as little as possible be eaten by taxes (amount to gift depends on what I receive, but for $2.5B probably a million each sibling (4) and 10-20 million to my folks)
4) Set aside a good amount for long-term living, so that even if I undertake anything and bottom out I still won't need to work at all
5) Find a nice area, probably somewhere 30-60 miles outside a major city, and settle down there (at least, use it as my "home base")
6) Donate vast swaths to select charities. EFF is up there, as is the UCLA. I also have more specific charities that I have personal relationships with, such as a poverty assistance center outside of Detroit, a feline sanctuary outside of Denver, and a cancer research fund that partners through my fraternity.
Once all of that is done I'll probably undertake a large number of economic experiments. Maybe something like MINCOME, but attempt to make it self-sufficient. I've always had a idea of taking the run-down areas of Detroit and turning it into a city of sorts for veterans, offering both housing and employment. These might be separate, or they might be the same. Such a city would also try to implement various "newer" technologies and be used for studying them, such as heavy solar installation and a revamped electrical grid.
Another notion I've long had is a website where people (likely just Americans) can apply for a $1000 grant for anything. Ideally I would partner with various companies to supply whatever was being purchased to help spread money further via sponsorship (and maybe a reality TV series), like if they applied for a grant to use towards a washer/dryer I would just buy the washer/dryer for them.
tl;dr: Pay off my debt, set aside money, they give most of the rest away