Hey [mention]Will709432[/mention]
Att få is a very nuanced verb, so it's not suprising learners of the the Swedish language struggle with it. I also feel like it is rarely explained other than through examples, which doesn't really help.
First of all, let's get some things out of the way.
Få can also bee an adjective meaning 'few'
Får, in addition to being the dictionary for of the verb, is also a noun, meaning 'sheep'
Now, on to the verb...
If you look upp att få in a dictionary or online translator, it will probably tell you it means 'to get'. And if you're lucky you'll get a list of other translations, but these won't help without context.
The general nuance of att få is to indicate 'recieving' an action from someone, if that makes sense.
Let's look at your examples, and try to exclude få:
Snart fick den tag i ett av mina rockskört = 'Soon, it got a hold on one of my coat-tails.'
Snart tog den tag i ett av mina rockskört = 'Soon, it grabbed a hold on one of my coat-tails.'
In this case, excluding få, switchs the focus from the fact that it grabbed the coat-tail, to the ACT of grabbing the coat-tail.
Folk slogs om att få komma in. = 'People fought to be let in.'
Folk slogs om att komma in. = 'People fought to get in.'
In this case, få adds the nuance that people were wight to BE ALLOWED to get in.
Jag lyckades övertala henne att få följa med en tjänare till stranden. = 'I managed to convince her to be allowed to go with a servant to the beach.'
Jag lyckades övertala henne att följa med en tjänare till stranden. = 'I managed to convince her to go with a servant to the beach.'
As you can see, without få, this meaning has quite a different meaning. With få the speaker convinces "her" to be allowed to go, while without få the speaker convinces "her" to go.
Han fick rätt = 'He was in the right.'
Han hade rätt = 'He was right.'
Again, this has the nuance of indication that "he" is recieving being right, rather than simply being right. The former is usually somthing that is officially decided, for example by winning a court case, and not commonly used in everyday language.
I'll make a list of some of the most common ways to use att få that I can think of. If you come across something that doesn't fit these uses, don't hesitate to ask.
att få = 'to get/recieve'.
Ex: Jag fick en cykel av mina föräldrar. = 'I got a bicycle from my parents.[/b]
att få tag i = 'to get/grab a hold of'.
Ex: Snart fick den tag i ett av mina rockskört och drog mig till sig = 'Soon, it got a hold of one of my coat-tails and pulled me towards them.'
att få <verb> = 'to be allowed to <verb>/to have to <verb>.'
Ex: Jag lyckades övertala henne att få följa med en tjänare till stranden. = 'I managed to convince her to be allowed to go with a servant to the beach.