Welcome! Always nice to see another commuter join the ranks!
You can do 1.5 miles on pretty much any bike. You can, in fact, do it reasonably quickly without a bike. So a few questions to help narrow things down a little bit:
1) Are you planning to do any other riding in addition to commuting?
2) What's your budget?
3) What's the parking situation like at your place of employment? Can you bring the bike inside?
4) Are you planning to ride only on nice days, or in bad weather as well?
Backpack will work fine at the beginning (just make sure it's waterproof if you do ride in the rain). It might be fine forever, for a commute that short. If you decide you want to switch to something else, you could always upgrade later (just be aware that some road/racing bikes might not have eyelets to attach a rack, which could limit your options).
Here are a few sample scenarios; it'll help you figure out which situation is closest to your own:
1) Are you planning to do any other riding in addition to commuting?
YES, LEISURELY MODERATELY LONG RECREATIONAL RIDES ON BIKE PATHS: Urban, comfort, or hybrid bike
YES, FASTER PACED AND/OR LONGISH RECREATIONAL RIDES: Flat bar road bike, cyclocross bike, touring bike
YES, FAST ROAD RIDES: Flat bar road bike, road bike, cyclocross bike, touring bike
YES, UNPAVED, SOMEWHAT TECHNICAL TRAILS: Mountain bike
NO: Any bike that looks good and feels comfortable. I'd go with the cheapest bike that satisfies those criteria
2) What's your budget*?
UNDER $100 - Used is your only option. Check out craigslist. You can get a serviceable beater for this price.
UNDER $350 - Used, craigslist. You might be able to get a decent hybrid/mtb for this price. New is possible, but pretty entry-level.
UNDER $500 - You can get an okay/decent bike shop bike for this price, or a nice used bike
UNDER $1000 - You can get a very good new hybrid/comfort/mtb, an entry-level (mind you, still pretty good) new road bike, or a really sweet used bike of any kind
ABOVE $1000 - Lucky you!
Lots of choice. Definitely an overkill for a 1.5 mile commute though, so make sure you do something else with that biek!
*Note that your budget should include a few things beyond the bike itself, namely:
- helmet (mandatory if there are helmet laws; otherwise, read the helmet threads to form an opinion)
- pump (pretty damn necessary, unless you already have one or live across a bike shop that has a 24 public-accessible pump)
- flat repair kit (though you could always walk your commute, and get a bike shop to repair your flats.. but doing it yourself is cheaper and generally faster)
- fenders (if you plan on riding when the ground is wet, you'll need them)
- rack and bike bags(optional, but you might want to go there eventually)
- lights (a must if you're riding within an hour of sunset or sunrise)
- multitool (comes in handy)
- lock(s) (how big and expensive will depend on how expensive your bike is and what the parking situation at work is like)
3) What's the parking situation like at your place of employment? Can you bring the bike inside?
SECURE INDOOR PARKING (preferably right by your side in an office) - Lucky you.
OUTDOOR PARKING - Two issues here: the climatic elements and the criminal element. The less secure and more exposed the parking area, the more conservative you want to be with the kind of bike you chain outside for the day and the better you want to lock it. How often do you expect to ride on rainy days? How is your area for crime?
4) Are you planning to ride only on nice days, or in bad weather as well?
This will help determine whether you need to bother with things like fenders, waterproof bags etc. Also, you don't want a really nice bike sitting outside in crummy weather a lot. In fact, if you want a nice bike and you want to commute in all weather and leave your commuter outside exposed to rain and vulnerable to theft, you might want to get two bikes: "the commuter" and "the nice bike".... But we all know where this is going... the n+1 bike fever. Once you start adding bikes to the stable it never ends.
Just be warned.