http://www.askdrsears.com/html/7/T070100.asp
Here's a great link for you, from Dr. Sears, per sleep problems in babies.
I would make sure, about HOW the Daycare puts babies to nap?
Do THEY let them cry it out? Do they comfort the baby? Do they pick them up if there is crying?
YOU as the Mom, should find out, and inform them that you do NOT want them to cry it out. Yes, for a young 3 month old baby... you should not do this.
Next, for his daytime naps... try and make everything dark, quiet, and calm. Do not do anything hectic or busy or too stimulating prior to his nap. For my kids for example, it takes me about 1/2 hour just to wind-down/calm them down for naps. This is called our 'pre-nap' routine.
BUT for a baby only 3 months old.. they can't just be trained or scheduled for naps, much less be regular about it.
For my kids, for example, it was not until about 6 months old that I introduced real regularity and routine to their naps. But, prior to that when they were younger, I KNEW when they were tired and needed a nap. I cued into them and them sleepy cues. I did not wait until they were 'over-tired' to then put them to nap. Over tired babies/children actually have a harder time to sleep and wake more.
Next, at 3 months old, they are changing developmentally and the awareness of their surroundings too, and it is a "growth-spurt" time where they need extra intake and get hungrier... which is in tandem with their changing development. THIS does put a kink into their sleep ability. It's normal. But not easy for the parent.
At this age though... they typically get tired after about 2-3 hours of 'awake' time. So look for your baby's sleep cues as well. ie: rubbing eyes, yawning, fist-ed hands, fussiness etc.
How many times a day do you try and get him to nap?
Being that he is only napping for 20-40 minutes.. he is probably still tired even after he wakes. This is a real short cat nap only... not a deep slumber which is restorative. Probably.
Each baby is different though.
Lots of babies hate being transferred to a crib after falling asleep and they do like just being in their parents arms. So its a hard transition... it will take time.. .maybe not until he is much older.... like at about 6+ months old.
Make sure you give him a good feeding before nap, burp him, change diaper, make everything calm. Maybe use some 'white noise" in the room for naps. (we put on a fan on low, when my son goes to nap. He likes it). Make the room nice and dark.
Also, a baby does not necessarily know how to self-soothe. Some do it no problem, others don't. My daughter was more high-needs we had to rock her & hold her etc., but my son just self-soothes on his own. Each baby is different!
there is no real 'formula' to this... but the link above is great and it will give you ideas and suggestions.
I don't know.. there is no one way... all Moms do things in various ways. Just some ideas on my part.
Good luck,
Susan