There are two essential responsibilities you'll have to take care of on your road to Financial Competence. One will apply to everyone; the other may not depending on your situation.
The first essential responsibility is to have an Emergency Fund -- a cash cushion to handle emergencies. These emergencies include house and car repair, unplanned medical expenses, or job loss. The key characteristic here is that these are unplanned expenses. You should be saving separately for planned expenses such as home remodeling, car purchases, or elective medical procedures.
Having this kind of emergency fund will keep you from going into debt when emergencies happen -- and debt is the enemy of financial security. (Debt can be a useful tool when you're fully financially competent, but until then should be treated like Kryptonite.)
The guideline for this emergency fund is that it should be 3 to 6 months of your living expenses. We're not there yet in the Schofield household. We've currently got about 1 month of expenses saved up. I expect the percentage of our emergency fund that we have saved up to increase quickly -- both because of increased savings and because our decreased living expenses will decrease the target.
The emergency fund should be kept somewhere it's safe, easily accessible (no penalties for withdrawal) and earns the most interest it can while being safe and accessible. The best choice would be FDIC-insured savings, but a money-market account such as Vanguard's Prime Money Market Fund would also qualify.
Currently our funds are in a Wells Fargo savings account earning 0.01%. (Yes, the decimal point there is in the right place. Sadly.). The Vanguard money market account is returning 0.03% right now, so it's not much better. I've found an online bank (Ally) currently paying 0.95%, and we're going to open up an account there. I've found that the best resource for finding a good savings account or money market account is bankrate.com.
The next essential responsibility is insurance, and there's several kinds you'll need.
In all of them, you'll pick as high a deductible as seems reasonable, because that does wonders to reduce the fee you'll pay for it. I'm going to do more research on exactly what levels of insurance you should get in the future.
Medical Insurance: If you can, get it through your job. Get as high a deductible as you can.
Auto Insurance: If you purchased a used car at a good price, (which is what you should have done) then get a level of liability insurance appropriate to your net worth and that covers the legal minimum required by your state. If you have a car with a higher value (which I do, but wish I didn't) then think about comprehensive and collision coverage (again, with a high deductible) and if you have a car loan (God forbid!) then you'll have whatever is required by your lender.
Home Insurance: If you own your own home, you should have homeowner's insurance. I don't, so I don't.
Renter's Insurance: Homeowner's insurance covers the home, not the contents. Renter's insurance fills that gap. If you have a high enough emergency fund, you might omit renter's insurance.
Life Insurance: If you have dependents, and you don't have enough investments to support them, you need life insurance. I have a daughter and plans for another, and a wife who's a full-time student, and our savings aren't anywhere near enough to support them. I need life insurance. I'll write more about this in the future as well.