In the last five years or so, I've been fortunate to go on several international trips. Even though it would be free to check a bag on those flights, I never do, and it has saved me so much time and trouble that I can't imagine doing it any other way. If you're doing any air travel, internationally or not, I'm here to tell you why and how you should avoid checking luggage.
Why to Avoid Checking Luggage
I'm an impatient person, and standing around waiting for my luggage to come around on a carousel is not my idea of a good time. It becomes extra annoying when I've just disembarked from a long international flight and all I want to do is sleep/eat/shower/collapse. And if you're a worrier like me, if you avoid checking luggage, it'll also save you from the anxiety that your stuff will disappear in transit (my sister had fancy chocolate stolen from her checked bags). Skipping the baggage arrival area and just walking out of the airport is surprisingly liberating, and definitely worth hauling a bag onto a plane.
If simple convenience isn't enough to convince you, the most important benefit I've found from living the carry-on-only-lifestyle is that it has saved me from missing connections. Exhausted from already having spent 17+ hours in transit, my husband and I arrived in Sydney, Australia later than scheduled, and were told we'd be missing our final flight. When the gate agent asked if we had checked luggage, though, and we told her we only had carry-ons, she said that if we just hurried over to the gate we'd have no problem. On the way home from this same trip, another delayed flight meant we would have missed our connection if we'd had to wait for our luggage to carry through US customs. (Speaking of customs, the fewer/smaller bags you travel with, the less hassle it generally is.) We've had similar situations on other flights since then.
Another plus to packing just what you can carry with you onto a plane is that it can make your transit cheaper and more convenient once you get to your destination. Your carry-ons will be easier to wrangle on public transport or on foot than a big, bulky checked suitcase, saving you the money you might otherwise have had to spend on cabs. In a big city with lots of traffic, this can be a considerable savings. Add in the money you'll save on checked bag fees over the years, and a nice compact rolling suitcase may pay for itself pretty quickly.
The bottom line: Checking luggage slows you down, makes you more likely to miss connections, and costs more.
How to Avoid Checking Luggage
Before you can pack your carry-ons, you need the right bags. You get two, one that fits in the overhead bins, and one that can fit on the floor in front of your seat. Choose wisely, and maximize them both. If you must bring a purse for daily use on the trip, make it a small one that can fit into one of those bags. Having tried a bunch of different combinations, I recommend investing in a good roller bag with spinner wheels for the overhead bin, and a backpack for down below.
Don't try to save money on cheaply-made versions, because you'll likely regret it and end up having to replace them. Buy the most sturdy, best-reviewed, comfortable-but-still-lightweight versions you can find. Sore muscles from hauling around a heavy, ill-fitting bag just aren't worth it (believe me, I did it too many times before I upgraded my gear).
If you want to maximize the amount of stuff you can bring, packing cubes are the way to go. The budget version is resealable plastic freezer bags. Your stuff stays organized, compressed, and with fewer wrinkles.
If I've helped motivate you to avoid checking luggage and pack lighter instead - but you're not sure how - I've written about some of the reasons why I used to over-pack, and how to overcome them. Curbly published Faith Provencher's tips for packing light, many of which are the same ones that I use. It might take a couple of trips to figure out what works best for you, so don't be too hard on yourself, and try to learn from each trip. Every time I travel, I make small tweaks, and realize that I need less stuff than I did the last time.