Apnea is an interesting area where you are doing things foreign to most people.
You can do things to promote the onset of the dive response (mammalian dive reflex) such as wetting your face with cool water - in particular the areas below your eyes (above the cheeks). Although some competitive freedivers with do dives with no warmup and "dry face" until they dive as this brings on the MDR a lot harder - IMO it is not much fun that way.
You can also train to increase your CO2 tolerance so that your body / mind is more used to doing things with higher levels of CO2 and not freaking out about it.
Static Apnea exercises can help - at least with your understanding of the feelings your body will go through. They don't all have to be "how long can I hold my breath for this time?" exercises. CO2 tables can help build CO2 tolerance e.g. a "pyramid" with static holds of a set time and ever decreasing surface interval in between each hold to a peak of 1 breath and then extending the surface time out again. O2 tables can also help - where essentially you have a decent surface interval in between each static hold - and e.g. do a hold until your first contraction, come up and breath and clear all the CO2 out, another hold until the 3rd contraction, etc.
What Barx describes above is essentially a variant of a CO2 table for dynamic apnea. You could also try a dynamic apnea pyramid. For starters do this with 25m swims and then change to 50m swims at another session. Swim the 25m (or 50m), have 1 minute breathing at the end, swim then 50 sec breathing, swim then 40 sec, swim then 30 sec, swim then 20 sec, swim the 10sec, swim then 1 breath, swim then 10 sec, swim then 20 sec, etc.
There are a lot of other exercises that will help you understand how your body feels. Never do any of these alone - always have an active spotter (not just someone sitting on the side of the pool).