What do you buy with them?
Pretty much everything. I probably average 2-3% cash back these days by having and using a few different credit cards that give high percentages back on different categories of merchants. One card I've been recommending a lot lately is the Sallie Mae MasterCard from Barclaycard. It gives 5% back on gas, groceries, and bookstores (Amazon counts as a bookstore). The bonus rate is good for the first $250 of gas, $250 of groceries, and $750 at bookstores in a given month. All other purchases get 1% back. If you're a Mustachian whose spending mostly goes toward essentials, you can't go too wrong by making this your one and only credit card.
I also carry a Chase Freedom card and a Discover card for their rotating 5% back categories, and an Amazon card for 2% back on restaurants and drug stores (but not Amazon purchases, since the Sallie Mae card gives a better cash back rate there). Though I happily carry multiple cards to optimize my cash back, you may want to start with just one card until you have established some good habits in paying off the balance on time every month. Remember that one month of interest charges can often be enough to outweigh a whole year of cash back on a card, so don't even try to play the cash back game until you know your processes for paying off your balance each month are rock solid.
How do you monitor your card so you don't have late payments (i.e. some kind of handy reminder system you use)?
All of my cards have an automatic payments feature that pays the entire statement balance on (or shortly before) the payment due date. The only risk in this system is that I might not have enough money in my checking account on the date the payment goes through. My card balances are usually low enough that this isn't an issue, but I try to remember to check the balance online every month just to make sure I put a little more in the checking account if the card balance was a bit higher that month.
Do you pay off the balance in full immediately upon getting a statement or in portions?
See above. I have the credit card company automatically take the full balance out of my checking account on the payment due date. Never pay less than the full statement balance. Therein lies madness.
Are there certain things (apart from just never paying) that I should avoid so I don't lower my credit score?
A history of paying your debts on time is the biggest factor in your credit score. Other factors include:
* Credit card utilization: it's fine to use your credit card for everyday purchases, but try to keep your overall credit card balance (across all your cards) below half of your total credit limit. Even lower is great if you can manage it. This shows that when you apply for a new loan, it's not just because you maxed out the rest of your cards.
* Age of accounts: Once you open a credit card account or two, never close it. Even if you never intend to use the card again, leave the account open. This helps keep your average account age as high as possible, which is a good factor in your credit score.
* Number of accounts: More total accounts on your history, with a good payment history, shows that more banks have trusted you with credit and you've paid as you agreed in each case. This also increases your score.
* Credit inquiries: Each time you apply for a new line of credit, it goes on your credit history for a couple of years. If you apply for a bunch of loans all at the same time, that's a small red flag.