When you check in for the first flight, you will be issued the boarding passes for all your flights. The boarding passes tell you what time your flights leave, what gate they leave from, and what time they board. No, they don't notify you that your flight is ready. You're supposed to read your boarding passes and show up at the time they say.
It's smart, if you aren't familiar with the airport you're connecting in, to go immediately to the gate your next flight is supposed to leave from, so you know where it is. At the gate your flight is supposed to leave from, there'll be a screen saying if it's on time or delayed or whatever. There are also screens here and there throughout the airport that give that info for all flights. If you go very far from the gate where your flight is leaving from, it's smart to check those screens from time to time in case anything has changed, but if you make sure to be at the departure gate at the boarding time given on your boarding pass and only then find out that it's now leaving from another gate, you still have at least 20 minutes to get to the new gate. They don't change gates very often but it does happen.