RY1 seems silly since S3 can perform its function directly, unless the issue is that you can't find a suitable switch to handle the current. A relay will be more expensive than just a switch of the same rating.
However, RY2 looks like a reasonable use of a relay. This is assuming the motors are not "frequently" switched on and off, like they would be with PWM control. Of course the relays need to be rated for the maximum motor current, which will occur when first turned on. This is often called the "stall current". See the motor datasheet.
The resistor in series with the RY2 coil doesn't make sense. From the labels, it appears you are driving the relay from 5 V. Get a relay with a 5 V coil. Most relays come in a family with the only difference being the coil voltage. 5 V is a common voltage, so you should not have trouble finding the right relay. 200 mA also sounds high unless this relay needs to handle 10s of A.
You might also consider a snubber across the relay contacts. That will extend life. If the motors are driven with DC and only one polarity, then a reverse diode across each motor will also help.