With due respect to Rob, who gave good advice, that approach should be a last resort. Since you have never done this before, it will be almost impossible for you to get your hole centered, and as you step up in bit sizes you will most likely destroy the threads before getting the bolt completely drilled out. Rob is correct in that you may in fact have to ultimately try to drill it all the way out, but take an interim step along the way per my post above.
The bolt is likely a grade 5, and possibly a grade 8. That will make it very difficult to drill, and a cheap bit will not do it.
First, heat it with a torch - MAP gas will work, propane will not - and let it cool. If the bolt is hardened, that will anneal it and make it possible to drill (thank
@WSS for that piece of advice). Then center punch it and drill a pilot hole - trying to keep it centered and straight as best you can.
This is a bolt extractor:
View attachment 261336
These are reverse threaded, and designed to be inserted into the hole you drilled in the bolt and tighten as you rotate counterclockwise. Use the biggest one you can fit in the bolt, and drill your hole out to the correct size for the extractor.
Heat it again, insert the extractor, and give it a shot.
If this does work, you will save the threads and not have to re-tap or re-drill and tap for a larger bolt.
If this does not work, then continue down Rob's path and drill it the rest of the way out.
To be clear, there is a risk here. Take care not to break off the tip of the extractor in the hole - they are hardened steal and you will not be able to drill it out with the broken tip in there. That's the reason to use the largest extractor you can for your sized bolt.
Hope that helps. Let's us know what you decide to do.
Jeff