As someone with asthma, I know how important the respiratory system is for overall health. Keeping your lungs and throat healthy and irritant free makes for easier movement, easier performance of activities, better focus, and just feeling better overall.
I cannot recommend a humidifier strongly enough. Aside from easing asthma and helping with allergies (by soothing irritated nasal passages and the throat), a study has shown that humidifiers can actually help prevent the spread of airborne viruses. They have also been demonstrated to help alleviate snoring. I find I am able to breathe more easily and feel like I am getting more oxygen as opposed to without one.
So many different kinds of plants can be beneficial to lung health. Green tea is a great anti-inflammatory, including for the lungs (add honey and/or lemon to soothe the throat as well). Licorice and eucalyptus are soothers for bronchial passages as well expectorants, which are substances (usually taken in the form of pills or syrups) to loosen mucus to drain the sinuses and make coughing more productive. Work to relieve a scratchy or irritated throat as soon as possible to prevent a cough or other issues.
From my perspective, sinuses are essential to full respiratory health, even though they are not technically part of the overall respiratory system. Keep on top of any sinus issues you may be expecting (headaches or dizziness, for example) to prevent infection. Keep nasal passages moist and clear by using saline sprays (available at almost all drug stores), decongestants, and expectorants when you feel you need them.
Most importantly, remember to consult with your doctor if you suspect you have a sinus infection or asthma. If needed, they can get you the medicine you need for your condition, or point you in the right direction of other resources if you have a different diagnosis for any trouble you may be experiencing.