Repairing electric baseboard heating

My home is about 40 years old (ranch style) and I have 2 baseboard
heaters that are not working. I have purchased new heaters, but need
to know how to attached them and how to troubleshoot what is the
problem. Heaters in other parts of the house work fine. I thought it
might be as simple as 1) shutting down the electricity 2) attacing the
wall wiring to the new baseboard 3) turn electricity back on and turn
the zone one and wait for the heat to come out! Anything wrong with
my plan of action?

Yes. Don't use use new parts as a test method. You didn't diagnose the
problem. Go back to square one. Open the breaker box. Use a test lamp
or voltmeter to determine if there is power at the terminal(s) of the
breaker supplying the heater(s). Be careful, use common sense there
and call a pro if you feel intimidated. If you have 240 V there the
breaker is likely OK. Remove the access cover on the heater and check
for voltage there. If present, your wiring is OK so return to the
service panel and turn off the breaker. Back to the heater, check for
voltage again to make sure you flipped the right breaker, and if none
present on either heater terminal, remove the connected wires. Check
the disconnected heater restance with an ohm meter and it will read
infinity if the element has burned out. If you read, say, 15 to 20
ohms, the heater is OK and the connections were bad. The usual clean,
tighten and so on would correct that problem. I assume your heaters
are 240 V and are supplied by a double pole breaker. Good luck.