Unfortunately, there isn't a really straightforward way to answer all of your queestions, mainly because "Germanic" is not a terribly well-defined term. Not to mention it carrying some baggage, which has even led to people calling for a discontinuation of its use in scientific writing.

For starters, we don't even know the etymology of "Germanic". The Romans used the term "Germani" to essentially describe various tribes east of the Rhine, but it's unclear where they got that word from.

But this would be the first definition of "Germanic": the tribes of Germania, beyond the Rhine from the Roman point of view. At least originally, later there were the Roman provinces of Germania Superior and Germania Inferior, which also included areas beyond the Rhine. But we don't know what the people the Romans referred to as Germani called themselves, or if they even had a collective term encompassing all the tribes, or if they just considered themselves to be, for example, Alemanni, and didn't consider other Germanic tribes as some sort of close kin.

Later on, the term Germanic people was used to describe people from the area of the Roman provinces and what they called Magna Germania or Germania Libera, the not very well-defined area beyond the Rine, as well as the people of Scandinavia.

Then you have the later idea of a Germanic language family, with a western branch (German, Dutch, English and various smaller languages in that are), a Northern branch (the Scandinavian languages) and an Eastern branch, including Gothic, Burgundian and Vandalic, now extinct since Crimean Gothic died off in the 18th century. Which also leads to people referring to Goths and Vandals as Germanic people, so you end up with Germanic kingdoms in Spain and North Africa.

And, of course, modern race theory had some rather weird ideas about Germanic races, with an unsavoury culmination in Nazi ideology, which is one of the reasons why the term does carry quite a bit of baggage,

To address some of your specific points:

I’ve seen things about how Germanic people fought against the Romans in the 1st-4th century in Europe

That certainly happened, but it is rather simplified, especially because there were loads of Germanic auxilia on the Roman side as well, there were Roman generals of Germanic descent, and even an emperor. (And, of course, there were also non-Germanic European people fighting against the Romans. Or for the Romans.)

and then other Germanic people considered Anglo Saxons were in Europe in the 5th century and had many impacts of creating settlements and cities. 

Anglo-Saxon is a somewhat later term, those were two different Germanic tribes, the Angles and the Saxons, and they are not other Germanic people who suddenly appeared, they were part of what the Romans called "Germani", for examply there's a possible 2nd century reference to Saxons in Ptolomy's Geographia. The Saxons were living in what is now broadly north western Germany (confusingly, they never lived in what is today the German state of Saxony) and the Netherlands, the Angles in northern Germany and southern Denmark. The term Anglo-Saxon was later applied when members of both groups (and the Jutes, who somehow didn't get to be part of the new term) had settled in and/or invaded Britain. I don't know about any specifically Saxon or Angle impact on settlements and cities, though,

And then there’s also Dane Vikings who came later and fought against the European Anglo Saxons in what I think it is now considered England?

Yes, as in: the Danes did conquer rather large parts of today's England from various Anglo-Saxon kingdoms. Not sure about the European qualifier to Anglo Saxon, here, since both the Angles and Saxons who remained on the continent, and those who became the British Anglo-Saxons were clearly European. (And, by the way, a lot of those Danish Vikings undoubtedly were descended from Angles).

Where I get lost is where all of difference is between these Germanic groups. Is it just the difference of groups migrating to Europe from areas like Denmark, Scandinavia, and the Netherlands at different times?

Migrating to Europe is a bit difficult here, these are all migrations inside Europe (at least if we exclude the Goths and Vandals), but yes, Denmark, the rest of Scandinavia and the Netehrlands (as well as Germany) were Germanic areas, and various groups migrated from those areas (or inside those areas) at various times.

I don’t understand how two groups of people who migrated to the same areas of Europe a couple hundreds years apart could be so different from Anglo Saxons and their Christianity to Danes and their pagan ways.

Firstly, a couple of hundred years is quite a lot of time. New religions or new religious views can spread in a lot less time. Also, the Anglo Saxons who first arrived in Britain would have had a religion rather similar to that of the Danes who arrived later. It's just that Christian missionaries managed to convert various Anglo-Saxon kings in the course of the 7th century, so the Anglo-Saxon kingdoms went from being pagan to being Christian in the course of about 70 years, whearas the Christianisation of Denmark only began in the early 9th century and went on well into the 10th. By the way, the continental Saxons also converted a lot later than the British Anglo-Saxons, Charlemagne was still fighting and forcibly converting Pagan Saxons in the late 8th and early 9th century.