First, you should NOT transmit on air bands as a UAS pilot. You do not have a ground station license and it would be a violation of both FCC and FAA rules, contact with the tower, when necessary, should be done via the phone. It is fine (and I'd even encourage) to monitor the air bands relevant to the location you are flying in to be aware of on-going traffic.
How to approach contacting the tower depends on what you are flying your sUAS as. LAANC (Low Altitude Authorization and Notification Capability) is now active almost everywhere, though some particular airports still do not use it. LAANC should be used whenever available, however the traditional approach will still need to be used for any particular airports not yet participating in LAANC.
If you are flying near an airport where LAANC is available, then you can simply use an app from your mobile phone to request flight authorization/notification. You simply enter the area or flight plan you will be flying and hit the button to request approval and then wait for the notification it's granted.
This can be done regardless of if you are a part 107 pilot or a recreational pilot. As long as you are flying below the maximum altitudes defined in the UAS Facilities Maps, you should receive near immediate approval in most cases. If you are flying above these limits, you will need to be a 107 pilot and go through the traditional waiver process.
You can find a listing of apps that currently support LAANC listed on the FAA's website here.
If the airport you are flying near does not support LAANC, then if you are flying for purely recreational purposes, you must provide notification to the airport operator and ATC prior to your flight. You should use the telephone contact information to do this. As long as they do not specifically deny your flight, you only have to provide notification.
If you are flying under part 107 (FAA certified remote pilot), you will need either an airspace authorization (easier to get and may be requested up to 6 months) or an airspace waiver (harder to get and may be requested for up to 2 years). You should check the UAS Facilities Maps to find the maximum easily allowable altitude for your flight and submit the airspace authorization request online through DroneZone.
It is currently taking around a week or two for authorizations that fall within the UAS Facility Maps guidelines and longer if additional research is needed to get approval. Authorizations are ONLY offered for airports not currently supporting LAANC as LAANC is the preferred means of processing.
One of the conditions of getting either a waiver or an airspace authorization is to provide details on how you intend to handle contact with ATC. Contacting the tower via phone is the most general approach. Some also monitor ATC frequencies so that ATC can more rapidly get ahold of them if needed.