Brake light and alternator light both come on The brake fluid level is ok and the battery seems to be charging.

DRIVE TO AUTOZONE,HAVE THEN TEST YOUR ALTERNATOR IT'S FEE,ALSO CHECK YOUR BRAKE FLUID,COULD BE LOW AND SETTING OFF SENSOR.

Brake Light. Make sure you are not driving with hand brake on. Check brake-fluid level. Top off brake fluid if low and make appointment to check brakes. Driving with brake light can be dangerous; deal with this soon. Call tow truck or drive carefully and slowly to shop.
ABS (antilock brake system) Light. Indicates either low brake-fluid level, stuck brake caliper or faulty ABS. Check brake fluid and make appointment to get brakes checked. Call tow truck or drive carefully and slowly to shop.
Oil Light. Either engine is very low on oil or there is no oil pressure. Pull over and turn off engine immediately. Check oil level and add oil if low. Turn engine back on to see if light has gone off. If oil light is still on, bad oil pump (no oil pressure) is likely. Call tow truck.
Check Engine Light. Car's computer senses engine problem. Difference in performance may not be noticeable, but make appointment to have car diagnosed. This usually cost around $65-$75 to do. OK to drive. If problem is serious, computer may switch to limp-home (low-power) mode and you won't have option of waiting to bring car into shop.
Air Bag Light. Get air bags checked. Problem could prevent them from activating in accident. OK to drive, but not safe in accident.
Battery Charge Light. Alternator is not charging. OK to drive, but turn off any unneeded electrical devices (radio, heater, defroster) and avoid starting engine more than necessary. Bring car into shop soon to avoid getting stranded. When alternator is not charging, battery loses charge and car stops working.
Emissions Lights (O2 sensor, EGR, check emissions). Depending on car model, these lights may indicate problem with emissions sensor or may light up at certain mileage, usually 60,000 miles (100,000 km). Get sensors checked. OK to drive. If problem exists, car might get poor gas mileage.
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The battery light indicates the charging system has a problem not a defective battery. as for the brake light that is unrelated and may be due to low brake fluid in the master cylinder. Have the Alternator tested to see if the output is within specs

When you replaced the battery, did you have the alternator tested? The battery light indicates either the battery is not holding a charge or not receiving the proper power from the alternator to charge the battery.
The brake lights could be a glitch but it could also mean that you are low on brake fluid or you have air in the brake system. Suggest you at least check the fluid level in the master cylinder and bleed the brakes.

It really sounds like maybe that alternator is bad. You can get it checked at a local parts house like an AutoZone, Advance or Pepboys. They can check the voltage at the battery with the car running, it should be above or around 14 volts, if it is lower then the alternator is not working properly. You should call and check to see if there is a warranty on this alternator if it went bad so quickly, but every shop and parts manufacturer has different warranties.
I can't say that that brake light is directly related to this but you should definitely check and make sure that the fluid level in the reservoir is at the proper level. The brake fluid reservoir is under the hood on the drivers side of the car right up by the firewall (the wall that separates you from the engine bay). It is usually a cylindrical container. You should see Max and Min fluid level marks. Make sure the fluid is in that range. If it is not you make need some brakes replaced. If the fluid level is fine it may be related to the fact that the battery is running low voltage, car computers get wacky when they do not get the proper voltage to them.

battery light indicates alternator is not charging. It may run fine until the battery is dead, but then you'll be stranded wherever it dies. Recommend taking alternator to parts store for testing. Once alternator is fixed, brake light may go out. If not, check fluid level in reservoir. If it's low, you may have a leak. Get back to me if you need help getting alternator out or brakes fixed, etc.

Wow. Good question. If you can get a hold of the schematic for your car you can check out the components that are in the path that turn on those two indicators. And since both are operating improperly at same time it has to be something that they both share in common.

OK, the brake light is caused most likely by the emergency brake cable being broken, and the pedal or brake lever is not in position.

Also, the brake light could be caused by low brake fluid level. Check that, too.

Battery light, is caused by the charging system not working, like a bad alternator or shorted out battery, or both.

I would have these issues repaired before driving any distance.

Checked fuses?