Don’t pack expensive, irreplaceable (as in jewelry), or electronic items in your luggage. I always carry my computer, camera, iPod, etc. in my carry-on. I know people who have had things stolen from their luggage while it was in transit. I also recommend getting some of those TSA locks and locking your bags closed. It’s not high security, but it makes it much harder for someone to just open up the bag and steal things.
Put a toothbrush, a change of underwear, any medications you might need, and so on in your carry-on. It really sucks to be in a foreign place and have the airline lose your luggage. Best case scenario, they get it to you the next day, but sometimes they don’t find your bags for days or weeks. You don’t want to be without your meds, for sure, and I personally would enjoy having something fresh to change into after traveling all day.
Be open, friendly, and patient when dealing with people. You know the saying ‘you catch more flies with honey than with vinegar’? It is so true, especially when traveling. People are so much more willing to help you out, or even to bend rules for you at times, if you treat them nicely and respectfully. It also opens you up to serendipity, as @marinelife said. If you’re open to possibilities, you may meet a local who has the skinny on what to do or where to eat. You could make a new friend or two.
That being said, exercise some caution, as well, but not in an anxious way, simply in a common sense kind of way. Keep an eye on who and what is going on around you, keep your belongings close at hand, and trust your gut if someone/something feels shady. I generally don’t let people bump into me, as skilled pickpockets use that as a diversion. Don’t keep all your cash in one place, in case your purse/wallet gets stolen, your pockets picked, etc. Keep only a reasonable amount out and available at a time, so you have a backup if anything happens to it.
Welcome to Fluther and happy travels! :)