My son wouldn't burp right away either sometimes. I would try patting him on the back or just bending him upright after feeding, which was the usual quick and easy way to get him to burp. But every now and then, he wouldn't so I'd give up. I'd just lay him in his bed and after a little while he'd wake up crying, uncomfortable, maybe in a little pain. Sure enough, I'd sit him upright just slightly and a burp would pop out. But, if he'd napped for about a 1/2 hour, it would cause him to then wake up and remain grouchy. Not good.
So, I got clever about it... because I wanted him to sleep for a long nap time and I didn't want him to experience discomfort. On the times that there was no immediate burp, I'd lay him down and then before he could even cry or get uncomfortable, I'd return to the room (just about 5-10 minutes later) sit him up slightly, give him a pat on the back, he'd burp and sleep soundly for a long time. I got to him before he could even cry; I got to him before the burp got to him. Interesting, huh, and simple... it worked.
As far as stroller rides goes - all kids are different. My boy was a big motion guy with a strong desire to suck so, an hour long stroller ride and a bottle to suck was pure heaven to him. Often he'd fall asleep in perfect peace.
But, both my girls have trained me that 15-20 minute car rides (or stroller rides - any strapped down activity) are all that they can handle before their eyeballs pop out of their heads. Pure torture (for them). We can only "handle" one errand (or outing) a day, errands (or outings) can't be several days in a row or rebelous meltdowns occurr, drive time to the occasion can only be 15-20 minutes long, play time must be a couple of hours long once we are there, and then a 15-20 minute return trip is barely tolerable. We stay home a lot too. And we live 5 minutes away from Walmart, the park, the library, close to town.
Speaking of grocery stores... now that's a challenge after the car ride to then also be strapped down in a grocery cart. Well, a corn dog is the only thing wonderful enough to my 15 month old that will allow me to borrow some of her time. So, yeah, it's tough. You might have to leave baby at home or not participate in strapped down activities for a while longer. Sorry :(