Infiniti G20 with ABS, Brake, and battery light on while driving.?

I let my 1999 G20 Infiniti sit for a week while on vacation and when I came back, my car had the ABS, Brake, Battery, and airbag lights were all on on the dashboard. I reset the codes so the airbag light went out, now I still have the ABS, Brake, and Battery lights glowing. The ABS light is only on when I am pressing on the gas and not sitting with the brakes on. I recently just got a new alternator, so I don't think that that is the problem. Could a bad battery cause this? Please let me know.