How to use 555 timers to control 2 relays
If you realy want a complex solution with 3 555 chips: read the datasheet, configure all 3  as one-shots with a common trigger, one for 10 seconds, second for 20 seconds, third for 80 seconds. Use the first one to activate the first relay (resistor-transistor-diode) AND to disable the second relay (you could use a spare contact of the first relay for that). Use the 2nd one to activate the 2nd relay (unless it is disabled). Use the last one to disable retriggering.

Or: learn to program a microcontroller, maybe an Arduino is a good fit for your level. 

If this is for a real product, any half-decent programmer can do this in any microcontroller, including the tiny 6-pins ones that look like a transistor and cost about half a dollar, like these: