When you first start a car it's common for all the dashboard lights to come on for a few seconds, but then disappear unless there's some sort of problem. The Brake light would indicate your emergency brake is on or that there's a fault in your braking system somewhere, and the battery light would indicate an alternator problem or low battery charge. The lights shouldn't go out unless the problem is resolved, so this is strange behavior.
What you need to do is get the codes read from the OBD port using a special reader. Any decent mechanic should be able to do this, or if you have the skills you can rent one and do it yourself. Either way the codes will tell you whether there's a problem with those systems or some sort of instrumentation issue.