Q: Alternator/Battery issues

Inasmuch as you installed a new battery, this is probably a charging system related issue, but the charging system includes not just the alternator but all interconnecting wiring. So, you do have to consider the possibility of high resistance connections, bad grounds and so forth. A quick diagnostic can be had by connecting a professional scan tool and measuring charging voltage (and retrieve any possible charging system trouble codes) during these episodes of the warning light coming on. If the charging output is out of spec., and the wiring tests good, that means the alternator is no good. I recommend that you buy a Toyota remanufactured alternator, Part Number 27060-0A050-84 for your car. Online from Toyota dealer part vendors, the Toyota alternator is around $309. If you do find the alternator is bad, I do NOT recommend you buy another "re-manufactured" one from a "parts" store, although brand new alternators are vended and if it is name brand, you could try that. Without a scan tool, you can check DC output at the battery (check for AC, too, as if AC is present, that itself is a sign of alternator failure) at engine idle and it should be roughly 14.2 volts measured on the battery terminals although the output voltage will be temperature dependent (exact values are in the Factory Service Manual). To have the alternator inspected and replaced, if need, please request an alternator inspection/replacement and the responding certified mechanic from YourMechanic will get this investigated and fully resolved for you. If you have further questions or concerns, do not hesitate to re-contact YourMechanic as we are always here to help you.