Bangkok is one of the most popular tourist destinations in the world, but also one of the most confusing and chaotic, especially for first-time visitors. I know this first-hand, as when I first arrived I was overwhelmed with the sight, sounds, traffic, street food, and general chaos. And like most first-time tourists I spend my first few days getting around by either tuk tuk or taxi. I then got used to the skytrain, but that was about it.
If you're going to visit Bangkok for the first time I'd definitely suggest you take at least one trip on a tuk tuk, as it's quite a fund experience roaring around the city on one. They are mainly aimed at tourists though, so the prices tend to be in line with what tourists will pay rather than local prices. As an example, some fixed price local buses cost as little as $0.20 for a journey, while a tuk tuk along the same route may cost a minimum of $5. So if you're on a budget, it could save you a lot of money by using local transport.
But having said that, you should know that Bangkok taxis are among the cheapest in the world, and it is worth paying a little extra to sit in a cab with aircon rather than in a hot bus, especially when you're stuck in traffic. But there are also aircon buses available, a rapid bus service on a dedicated road, the underground and the skytrain. All these have great aircon, but often too cold for locals.
Then there are the less well-know ways to travel the city that most tourists never find out about. These include motorbike taxis, bike shares, songthaews (a cross between a small van and bus), and even boat.You really should check some of these out, as they can be a fun and cheap way to get around the city. There are at least 11 ways to get around Bangkok, although I found another two that aren't on that list.
The first is the airport rail link that connects Bangkok's main international airport to the skytrain system, and also vans - basically minibuses that travel mainly from the center to the outskirts of Bangkok. These are great for day trips to places like the floating flowing market. You'll find many of these vans neat Victory Monument skytrain station. Just ask around for specific destinations. It seems like you can always find a van going to your specific destination.
Bangkok is a very friendly city, so get out and about using local transport and meet some local people. Learn even a few words of basic Thai and the locals will love you for making the effort. You'll really feel at home and will get some great tips for things to see and do that you might otherwise miss.