Most of the following should be obvious, but they haven't been so obvious to myself and some of my friends in the past...

General Travel Advice

Don't carry more than two bags. You have two hands so having more than two bags means more than two bags in one hand, which makes you a good target for pick-pockets. Also if you have two bags then you can probably control them both with one hand while buying train tickets etc (a good time to get pick-pocketed).
The only reason I can think of for having more than two bags is when travelling by plane in the US (where the limit is two bags in checked luggage and there is no limit on weight). Otherwise one bag that's checked and one for hand luggage is safest.
Avoid glass bottles as much as possible. Not only is there risk of breakage when travelling, but there is more risk when you reach your destination. I find that I'm more likely to drop a bottle of deodorant or after-shave in a foreign bathroom than I am when at home.

Airline Advice

Avoid carrying liquids in planes wherever possible. Also never have a liquid in your checked luggage (checked luggage has lower pressure, gets dropped more, and you can't re-arrange it in-flight if something breaks). Alcohol is particularly risky as it will rapidly soak through clothes etc, but even shampoo can cause a surprising amount of damage.
The only liquid I will have in my checked luggage is toothpaste (it's too thick to spread far and tends not to damage anything).
I recommend checking bottles of alcohol once the plane is in the air, I once noticed that a bottle of bourbon I was carrying had a broken seal - fortunately I hadn't lost much by the time I noticed and tightened the lid so the rest of my luggage was not damaged.
Always ask for a window seat! The reason is that most people ask for a window seat, so if you have a window seat the probability of the seat next to you being empty is greater. If you're scared of flying then get over it and take a window seat.