First time setting up BGP and I'm sure this is something simple.

I've successfully peered my MikroTik router to my existing core router. All 480,000-ish routes have poured in - it's talking just fine. That said, I can't even ping the outside from the terminal in Winbox. I can ping my DNS servers and my other internal devices but cannot get to 4.2.2.2 or ping google.com - it times out. I'm convinced there's one little button I forgot to mark.

Thanks for any ideas.

assuming your routers are within one ASN, did you check route reflection on their peering session on both routers? Also, did you enable IGP protocol such as OSPF?

Not using OSFP and both routers are getting routes from each other.

From the mikrotik terminal, I can resolve domains' IP addresses, but pings time out.

Traceroutes get to my immediate bgp peer and then fail.

In your MikroTik router, go into ip routes and do a search where your dst-address is 4.0.0.0/9 (your 4.2.2.2 test).
If your "Gateway" (next hop) isn't your core router-- you need to enable "nexthop-choice=force-self" on your core router's peer configuration for your MikroTik. If your core router is Cisco, it's probably something like "neighbor next-hop-self".

Thanks for your help,
Looks like this issue was caused by some misinformation from our network administrator. Once I got the proper settings I was golden.