Not to be a pessimist, but you might want to look into a business degree with a math minor. You'll come out of undergrad and earn some decent money, enough so that you can afford to put yourself through grad school. I was a hard science major until I realized my options would be so limited and I'd be mostly broke, so I went the route of a BBA in Economics with a whole lot of math credits that didn't add up to a minor (too much statistics, didn't count). I'm a bit over a year out and am making enough that I can afford to live comfortably and go back to school at night. You could always minor in physics instead of math if that's what you like. It'll make your life a lot easier if you can come out of undergrad with good earning potential.
Or, take your second option, and go the engineering route. You'd see a good starting salary with mech/chem/electrical eng, once again enabling you to pursue a graduate degree of your choice.
I'm not trying to tell you what to do, nor am I trying to bash on programs/majors/careers. I'm just laying out my own situation and how it's working out so far.
By the way, career questions are often asked in the Fatwallet Finance forum. Check it out, there's a lot of good stuff there.