You don't get a credit score for free if you go to the real credit report website:
www.annualcreditreport.com.
If you do get a free score, there is going to be 'e-baggage' whether it be giving your credit card only to find out that you didn't read fine print and you signed up for $19/month credit monitoring. Or you get an influx of spam to your email address etc.
750+ is solid.
Your use of credit cards sounds very good....using a CC transactionally and not revolving is a very good practice. However, something that I learned from a friend is that even if you charge everything to a card and then pay it off, that you should still not try to exceed 20% of the limit on the card. This can be hard to accomplish unless you have a card with a high limit. I'm pretty anal about my credit, so I even go as far as to sending a payment the day before the end of the cycle so that my next statement balance is only a few bucks (representing any pending charges from that day). This way, on my credit report it shows that my balance is only $32 or $45 etc as opposed to $2500 or $3000. Granted, even if you your balance is a few thousand and you pay it off right away before the due date, you don't pay interest, but it will show as a balance nonetheless on your credit report.
I would not accelerate the payoff of your student loans. However, if you can, getting the car paid off will be good. It's thought of as consumer debt which carries a slightly more negative connotation than student debt (but no debt is good debt

). This has always bothered me because at least a car loan is secured by something of some value (granted it is a divesting value) whereas college debt is unsecured...alas.
Also, try to calculate your DTI to be about 36% to get an idea of the mortgage payment you could afford (read: get approved for):
(sum of loan payments per month + hypothetical mortgage payment per month)/gross income = .36
or
hypothetical mortgage monthly payment = .36* gross income - sum of loan payments
Banks are including escrow payments in that calculation as well nowadays so that payment represents the sum of: principal/interest, hazard insurance per month and property tax per month.
Another thing to mention is that the downpayment is getting very very important nowadays. My neighbor wanted to keep some liquid savings so he only wanted to put 10% down and the bank said no and he had a 810 credit score (experian)! So with that said, maybe not paying down your loans would be better if it allows you to put more down on the mortgage.
If the monthly payments are really high but the balance is relatively low, paying it off can be really good for your DTI. If the monthly payments are low then it doesn't adversely affect your DTI so it may not be worth paying off early. Granted DTI is not the only metric that creditors use, but generally a good easily derivable metric to figure out your situation.