For drilling glass, drill bits for tiles (the spear-point type) work. Work wet and go slowly. I've seen them down to 3mm. You need to hold the glass steady, but a cordless drill will be adequate.
If you want a smooth (as opposed to facetted) piece, jewellers rouge on a cloth will polish glass, worked by hand, though it might take a while. For a little reshaping, such as to remove a sharp edge, you can use wet/dry paper (silicon carbide paper) or emory cloth. Start coarse and work down to a fine grade before polishing. I'm much more familiar with machine polishing, but that's good for flat surfaces.
If you really need to cut it, small cuts can be made with a dremel or similar, with a carbide cutoff wheel. There are also coarse hand saws for tiles that cut glass. They're like a file made of thick wire (coated in abrasive) in a frame, rather than what you'd it think of as a saw.
Safety glasses are a good idea when cutting or drilling.
First you need a tool to work the glass. I suggest a dremel or similar rotary tool for that, but there are vast differences in quality:
- You can get a dremel / rotary tool including a collection of bits and a tool box for around 40 €/$. Most of these advertise insane rpm values, but in reality what you want is motor power, not motor speed (I never used more than half the available speed of my tool). They notably slow down as soon as your bit touches the object because the motor doesn't have the power to withstand the slightest resistance, forcing you to work in very small increments. These cheap tools are okay for occasional hobby crafters, but if you plan on working with them regularily, you'll be very dissappointed.
- Many people already have such a tool and never use it. Maybe you can loan one from your neighbor / friend / family for a while.
- If this is more than just a one-time project you should consider getting a more expensive, high quality tool. These have a strong motor that allows you to work fast and evenly and most include nice features like a quick and secure way to change bits or an ergonomical handle.
Then you need the bits to use with your tool. I'll refer you to Chris' answer for that with the additional advice to totally forget each and every cheap bit set that is included in a cheap rotary tool. In my experience they are not properly centered, which makes them bump into the object instead of running smothly over it, causing damage or rough surfaces where you wanted to smoth them.
For shaping the glass and drilling holes you could ask your dentist for old drill bits they no longer use. They won't be perfect (after all your dentist sorted them out for a reason), but it's a cheap way to get good diamond bits for drilling small holes.
Then you absolutely need protective equipment, at least safety glasses and a fine dust mask! The fine dust you produce while working the glass is not as smooth as it feels. It's a pile of microscopic glass shards that can irritate your eyes and lungs. Even worse! Since glass isn't organic, your body cannot decompose it and it stays in your lungs forever.
And lastly you need patience, lots of it. Polishing glass to a mirror finish takes several passes with decreasing grain size and finishing with polishing paste. To me personally it sounds not worth the efford for an oddly shaped piece of a bottle. Many people like the frosted finish of beach glass.