I would like to start experimenting with Robots. Is Lego Mindstorm a good start? Should I consider other platforms?
If you want to focus on the software only, then Mindstorms are a great start. Because you have a fixed platform and do not need to care a lot about the hardware. And if you want to mount some homebrew sensors this is also possible. The mindstorms itself are well documented and the hardware/software interfaces are relatively easy.
You can do some stuff by using the Lego-NXT Programming IDE or even write your own programms and let them run directly. At the end you can even upload your own OS to the Arm7 and do whatever you want. Just do not forget the Samba (see NXT Hardware SDK) If you have more question to the Lego-Robots, there is SE for this, too. https://bricks.stackexchange.com/
Instead of using Lego you can think about buying some Arduino/Raspy based robots, where you might need to get your hands more dirty (from hardware point of view)
I played quite a lot with the lego mindstorms and indeed they are interesting but I found them quite limited at the ARM level. The other point with them is that when you move to another Core like the raspberry which I find more interesting (but I am a developper) is that you can continue using the lego parts with it.
So depending where you want to focus on, lego is a good start for all the bricks, sensors and motors but if it is for more programming raspberry will be a good choice.
Tosbiak gave some link on the lego so I will just complete with a link on raspberry: https://www.raspberrypi.org/resources/ (for exemple the butler is quite and interesting robot to make)
In addition to Mindstorms being a good start for software, there are competitions for younger students such as FIRST robotics, and Botball where Legos plays a large role(may not be found everywhere though). You learn software, and also how to design a robot to perform a specific task or tasks.
However you will not learn the electronics side thus missing out on a huge part of the experience. While I haven't tested it, I started with a kit similar to this Parallax BOEBot from Adafruit. It serves as not only an introduction to software, but also the parts involved and assembly. The downside here is that you're not working with Legos and so the ability to reconfigure and build something new is lacking.