Agree with Dan.
The key questions are:
# how much do you want to invest (do you have enough to invest in ~20 companies)
# can you afford to lose it
# what is your current basis of knowledge.
# how much time do you have to learn
If you have time and money, go EIR for a vc/angel for free (have a friend who did this) and learn for a year.
If you only have $10k, don't bother.
Short answer: read Tony Robbins' new book, Money: Master the Game. It isn't as kitschy as it sounds, and is everything you'll need to know, from the ground up, from a man who has set out passionately to help everyone at *all* levels of investing, from pennies to billions.
As exciting as it might be to invest in a startup, that isn't prudent unless you have a lot of experience evaluating business models and running successful companies. There's a real danger of falling in love with someone's airy idea (which is fun) but losing your entire $10k when their startup fails to find its footing and goes under.
If you had very specialized knowledge of some market sector such as real estate or the domain name market as well as time to spend researching opportunities, then you might invest in land or houses or domains or whatever.
But the barrier of entry for buying real estate – at least for a solo buyer – is above $10k. Meanwhile, the domain industry is much riskier; and people tend to spend their first year losing money in order to learn. So I don't recommend that unless you have a year to spend studying a very peculiar market.
It may sound dull, but I'd recommend diversifying your investments as much as possible and buying into very safe asset classes. You can divide $10k across a handful of mutual funds at Vanguard.com. Each of those mutual funds would spread your investment across numerous stocks. Pick your flavor.
That's what I did for a decade, branching out into individual stocks and a home midway through. You can earn a decent return that way while limiting the damage from a worst-case scenario. Mutual funds are largely a hands-off investment, yet you can study the actual companies the fund is invested in as much as you wish to.
Mutual funds are managed by professionals who spend more time than you can researching the companies themselves and watching market conditions. While I was underwater (not with a mortgage, with a boat), I literally couldn't pay attention to the stock market for months at a time, which made investing in more volatile individual stocks a bad idea.
Mutual funds provide a good balance of growth potential and safety. Meanwhile, you can read up on the riskier areas of investment. You'd be able to withdraw your funds whenever you choose. Except – that is – if you're investing in a tax-protected vehicle such as a Roth IRA or 401k (for U.S. investors). You ought to place investments within those plans for tax purposes. Any financial analyst would say so. However, you'd have to leave the money there until retirement time to enjoy the tax benefits.
If you're young and drawing a salary, then you can afford to lose money, knowing that you have years ahead of you to recover your losses. Later in life, we don't have as much time to offset bad investments; so most of us choose safer investment vehicles as we age.
In general, though, invest in the safest areas first – those that demand the least involvement from you. As you learn more and can put more time into the research, branch out.