First look at your patient - is the airway PATENT? how are they breathing? check your vent settings. know your pressure alarms. Typically high pressure alarms are due to coughing or a blockage (kinked tube, mucous plug). Low pressure alarms/low exhaled tidal volume alarms are due to dislodged tubing or a tubing/ETT (cuff) leak. Always check your tubing.
A high respiratory rate alarm could be due to a kink in the tubing or condensation that has built up in the tubing. The patient could be in respiratory distress, too so also look at your settings. If the vent rate is set at 10 and your patient is breathing 40 times/minute, they could be in distress! That may be due to inadequate sedation, too.
Apnea alarms are usually due to the tubing being disconnected, or the patient has actually stopped breathing.
Correlate your vent settings with the ABG. Respiratory therapists are your friend