Sorry it took some time to reply, but as a trial member, I can only once a day post, and only for 6 days, I think.
Thank you for explaining the English. To me "went running to the park" sounds like "running" is just a modifier telling HOW they went to the park.
Maybe they were just in a hurry coz they wanted to meet someone there and are late? How did the go there? Running! vs. the went to the park in order to run (there).
I may be totally mistaken though, my English ain't very good. Or maybe it depends on the actual context. maybe it's all the same. dunno.
getting back to your original question:
>>In the beginning lessons though, we learn phrases like "estoy aprendiendo" = "I am learning." It follows that "I went running" would be "fui corriendo" in that scheme.<<
Note that 'estoy' is a form of 'estar' whereas 'fui' is 'ir'. Grammatically they do not assume the same roles, hence both structres do NOT match here.
You don't build these progressive forms with 'ir' + gerund but with 'estar' + gerund. Just like "I AM learning" is not the same as "I went running."
The parallelism would rather be "I was running" (Estuve/Estaba corriendo).
As in my above post's example "fui corriendo" no significa lo mismo que "fui a correr", sino que expresa la manera de que me fui. Here pretty much like "I ran (there)".
If you add 'me' you get "Me fui corriendo" - "I ran away".
Besides that, "fui a correr" and "salí a correr" means about the same. literally, one is just went the other went out, so of course, you canot use the went out version
if you have not been at home o en un sitio del cual has salido. Tiene sentido, ¿no? :)
So you can use "Las chicas salieron a correr" y "Las chicas (se) fueron a correr".