I assume you mean it is 11 miles each way? 22 miles each way is something like an 80-minute commute twice a day, possibly.
Eleven miles each way is just enough to get warmed up for the work day--perfect.
The questions are---what do you need to carry with you? And ... is there a safe place to park?
For an 11-mile ride, any bike which isn't broken should be fine. You are thinking properly that you should get a cheap bike and learn about what really works for you. Any brand .... except maybe department-store bikes--should be fine. Mostly what would matter for the money you are planning to spend is that the bike Fits .... most important .... and then, that everything works. Make sure everything that should spin, spins, and doesn't wobble or grind.
If you don't know much about bikes, you definitely want someone who does to go with you to check out prospective purchases.
For an 11-mile ride, unless you are riding off-road, I would suggest a rigid frame endurance-geometry bike with rack mounts, making sure it has clearance fore tires at least 32 mm wide. Flat-bar bikes can be good for short urban commutes because you won't be spending long periods of time in an aero position, and being able to see well at intersections can be a survival issue, but I have used flat- and drop-bar bikes successfully. The big issues for urban commuting (in my experience) are excellent lights, good brakes, pretty healthy tires (as in wider and higher volume, for all the broken pavement, rail crossings, pavement seams, and the occasional unavoidable curb hop or drop,) strong wheels (same reason,) a rack which can hold all your gear, bags which are completely waterproof--or learn to pack water-proof----and a wide range of gearing, because there will be days (particularly towards the end of the week) when your legs will feel like someone has been hitting them with a ball-peen hammer, and having low gears can save you.
I would recommend either and aluminum frame and carbon fork, or a steel frame and fork, fir the bike you ultimately buy. Carbon frames can be lighter, but in my experience there will be people in the office who decide to move your bike and drop it, or bump it, and urban riding occasionally leads to slips or spills, and sometimes you might have to lock up in odd places. an aluminum or steel frame might get a little scraped but nothing will break it (which doesn't break you first) whereas a CF frame might take damage which will be invisible until it fails in a big way. (Just fr urban commuting, which can---but isn't necessarily---a pretty abusive sport. it depends on your roads,, your route, your office---the is a guy here, @
Jim from Boston, who rides a high-end CF Wunderbike through some city...I forget which
---several days a week with no ill effects.)
For your first bike, anything that can fit a rack will be fine. Just make sure that all the parts work. You will spend more time learning about cycling, and how you ride, than about the bike, and after several months when you go to bike stopres and web sites you will know what matters to you and be able to make wise choices.