On average, you should be saving at least 10-15% of your income in order to be financially secure when you retire. Different people will tell you different things, but really this can be split between short term savings (cash), long term savings (401ks, IRAs, stocks & bonds), and paying down debt.
That $5k is a good start on an emergency fund, but you probably want a little more. As justkt said, 6 months' worth is what you want to aim for. Put this in a Money Market account, where you'll earn a little more interest but won't be penalized from withdrawing it when its needed (you may have to live off it, after all).
Beyond that, I would split things up; if possible, have payroll deductions going to a broker (sharebuilder is a good one to start with if you can't spare much change), as well as an IRA at a bank.
Set up a separate checking account just for rent and utilities, put a month's worth of cash in there, and have another payroll deduction that covers your living expenses + maybe 5% put in there automatically. Then, set up automatic bill payments, so you don't even have to think about it. Check it once a month to make sure there aren't any surprises.
Pay off your credit cards every month. These are, by far, the most expensive forms of credit that most people have. You shouldn't be financing large purchases with them (you'll get better rates by taking a personal loan from a bank).
Set specific goals for savings, and set up automatic payroll deductions to work towards them. Especially for buying a house; most responsible lenders will ask for 20% down. In today's market, that means you need to write a check for $40k or $50k. While it's tempting to finance up to 100% of the property value, it's also risky considering how volatile markets can be. You don't want to end up owing more on the property than it's worth two years down the road.
If you find yourself at the end of the month with an extra $50 or so, consider your savings goals or your current debt instead of blowing it on a toy. Especially if you have long term debt (high balance credit cards, vehicle or property loans), applying that money directly to principal can save you months (or years) paying it back, and hundreds or thousands of dollars of interest (all depending on the details of the loan, of course).
Above all, have fun with it :) Think of your personal net worth as you do your Gamer score on the XBox, and look for ways to maximize it with a minimum of effort or investment on your part! Investing in yourself and your future can be incredibly rewarding emotionally :)
