As far a I know, it's a bit inaccurate to think of Spain having always been "Spanish". Even today, there are many different languages and cultural subgroups like in Valencia, Catalunya, Andalusia, Galicia, the Basque region, etc other than the "standard" Castillian culture and language. One can assume that back during the colonial era, these differences were even greater. So my question is, did the regions settled by Spain become influenced in culture by those who came over from Spain, and can we see these influences today? Or were most colonists Castillian in origin or am I totally totally off base about Spanish culture(s)?
Yes, during the colonial region that differences between the Spanish regions were as great as they are today or perhaps even greater.
So my question is, did the regions settled by Spain become influenced in culture by those who came over from Spain, and can we see these influences today?
In a way, yes. It is difficult to assess (although there are probably some studies about this particular topic), but some influences are noticeable. For instance, during the times of the Spanish Empire, the port of Seville, in Andalusia, was awarded royal monopoly of all commerce, transport and basically any form of contact with the colonies in the Americas. That meant that all ships that went and came had to depart from and arrive to Seville. Andalusian culture has had as a consequence a meaningful impact on the "Spanish" culture of the Americas, and alongside Castilian culture is probably the one with greatest influence in the new world (others like Catalan, Aragonese and Basque have had minimal impact).
The Spanish colonial architecture, although highly variable from region to region and with local American influences, has many identifiable Castilian and Andalusian traits. On the other hand Basque architecture or Aragonese architecture (which are quite distinct) have had a minor impact.
The Andalusian dialect of Spanish is also believed to have been the basis for the current forms of Latin American Spanish, as it is very distinct from standard Castilian Spanish, lacks the /θ/ sound (replaced by the /s/ sound) and prefers the use of ustedes to vosotros for the 2nd person plural.
Or were most colonists Castillian in origin or am I totally totally off base about Spanish culture(s)?
Most were Castilians and Andalusians, but bear in mind that Andalusians are basically Castilians that repopulated the Al-Andalus region after the Reconquista. That's why in the Hispanic population of the Americas surnames of Castilian origin (Fernández, Pérez, González, Suárez, etc.) are amazingly frequent while other surnames such as Basque, Galician or Catalan are much more uncommon, Catalan especially.
There were not American regions settled by specific Spanish populations (such as a city settled mostly by Basques, another one by Andalusians, etc.), most were mixed but with a higher number of Castilians and Andalusians.
A lot of people from South America (the huge majority, I'd say) speak castilian with seseo (/phonemes /s/ and /θ/ are both pronounced as [s]) because most of the colonialist/inmmigrants were from south Spain or Canarias.
Sadly, bullfighting in Mexico and other places comes from Spain.
A lot of people from all South America share basque and castilian surnames too.
Gastronomy, folklore, sayings, names... there are small examples all over the place.
These cultures are as spanish as the castilian culture, though. In the same lane of thought, the bavarian culture is also a german culture; even if the bavarian isn't german, but germanic. Castilian is only equippable to spanish if we're speaking about the language.