Robotic courses

I can kind of relate to your question and will try to answer it based on what I've learnt in the past few months. I'm a mechanical engineer and a few months back, I had an IT related job with no skills in Robotics except maybe a simple simulation or two. But I really wanted to pursue the field, so I quit my job and started a Masters degree in Robotics from the same institute I did my Bachelor's from. My initial plan was to gain some knowledge in Robotics and apply for a Masters from a better ranked university abroad for that. However, over the past few months, I've figured that I could have done that (or better) by continuing my job (and earning something), and self teach myself Robotics along with that.
There's really a simple way: MOOCs or online courses. Since Robotics is one of leading research areas nowadays, a lot of material is freely available on the internet for self study. Here are some of the MOOCs I took which have really taught me a lot about Robotics:
1. SNUx Introduction to Robotics courses part 1 and 2 on edX. These will teach you about the mechanics of Robots.
2. Control of Mobile Robots by GeorgiaTech on Coursera.
3. Autonomous Mobile Robots by ETH Zurich on edX. One of the best Robotics related courses available.
4. Introduction to Robotics by QUT.
5. Robotic Vision by QUT.
6. Cognitive Neuroscience Robotics by Osaka University on edX
7. Machine Learning by Stanford University on Coursera.
8. Underactuated Robotics by MIT on edX.
9. Artificial Intelligence by UC Berkeley on edX.
10. Autonomous Navigation for Flying Robots by TUM on edX.
11. Robotics specialization by University of Pennsylvania on Coursera.
12. Mobile Robots and Autonomous Vehicles by INRIA on FUN-MOOC.
13. AI Programming for Robotics by Google (Sebastian Thrun) on Udacity. (Highly recommended)
14. Machine Learning course videos by Tom Mitchell (as taught by him at CMU)
15. Machine Learning course videos by Andrew Ng (as taught by him at Stanford)
16. Introduction to Machine Learning by Sebastian Thrun on Udacity.
17. Machine Learning specialization by University of Washington on Coursera.
18. Introduction to Computer Vision on Udacity.
19. NPTEL Introduction to Robotics (Thank you Shaham for the suggestion).
20. Computational Probability and Inference on edX (Useful for AI, ML and SLAM related concepts).
21. Neural Networks for Machine Learning by University of Toronto on Coursera (One of the most recommended starting resources for people wanting to explore Deep Learning).
22. Artificial Intelligence Planning by University of Edinburgh on Coursera.

These are some of the online courses I have taken, am currently taking, or plan to take soon. These will teach you a lot of Robotics and with the knowledge gained from these courses, you can possibly think of pursuing research on your own in your interest areas.

Additionally, make a hardware robot yourself and experiment on it, or simulate one. You also need to have good programming skills in C++, Python, and MatLab which are the most commonly used languages in Robotics I think.

After you have gained some knowledge, it would be really beneficial to learn ROS (Robot Operating System). It's one of the most popular emerging Robotics development platform and there are excellent tutorials available online to help you get started.

I hope this helps you get started with Robotics. It's an awesome field and there are so many open questions still left to answer and so many challenges still to solve, and I love it. I've learnt more from these than from my graduate courses, and think you can do the same. It will definitely add value to your grad school applications, and you would already know so much about what they will teach you there, and help you better concentrate on projects and research there.

MIT
Mobile Autonomous Systems Laboratory