1 - "I picked up my phone and called up my friend".
2 - "I went up and got myself registered".
3 - "Early in the morning we got up"
I never know when to use the term "up".
How would these sentences would be without the word "up"?
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The key is to understand phrasal verbs, which are a combination of a couple words to give the phrase a different meaning than the individual words spoken consecutively. These phrasal verbs are typically listed in a dictionary, there's no magic way to learn them, except one-by-one.
For example, the phrasal verb go up has two meanings: to be engulfed in flames ("After the fire started, the house went up in no time."), or to be built, as a building ("After the fire, the new house went up in no time."). The first example means the house burned up very quickly, the second example means the replacement home was built very quickly.
You can also say "Jack and Jill went up the hill," but that's not a phrasal verb – it simply means the two people moved upward.
To show how tricky this can be, consider these, where the word up can be removed with minimal change in meaning:
We need to burn up the leaves. vs. We need to burn the leaves.
Ed tried to stand up on the table. vs. Ed tried to stand on the table.
We should fill up the gas tank before we leave. vs. We should fill the tank before we leave.
Hurry up – drink up your milk. vs. Hurry – drink your milk.
However, in these sentences, the up is absolutely crucial:
I hope the rain will let up by this evening. (not, I hope the rain will let by this evening.)
My uncle will stand up a new branch next month. (not, My uncle will stand a new branch next month.)
I'm not sure where I'll go; it's a toss up right now. (not, I'm not sure where I'll go; it's a toss right now.)
With that in mind, here are the ones you asked about:
1 - I picked up my phone and called up my friend.
Here, the first up is required is the second up is extraneous. To pick up means to lift, but to pick means to select. So, you wouldn't say "I picked my phone" unless there were three phones to choose from. On the other hand, call up is also a phrasal verb, but it means to telephone – and so does call. So the second part of the sentence means the same thing either way; but I'd probably eliminate that word.
2 - I went up and got myself registered.
The up should be removed from this sentence. (In fact, it should probably be reworded to "I went and registered.") As mentioned earlier, went up means to have been built or to have been burned; neither of those fits here.
3 - Early in the morning we got up.
The up here is both acceptable and necessary. The verb get means to obtain; the phrasal verb get up means to awake and arise from sleep or to get out of bed. There are two other ways you can say this; these alternatives show how tricky it can be, knowing when to use the word up:
We woke up early in the morning.
We awoke early in the morning.
The word up can be paired with the verb wake, but not with awaken.
There's no easy rule that would tell you when to include or omit an up; you simply have to learn the phrasal verbs. A good number of them will be listed in a dictionary (under the entry for up), but it's not that easy, because others will be listed under the word up is paired with (such as call up, which can be found at that link, under Definition #66, where it says "See also", right after call off and call out). If I were to try to list them all, I think that list would build up pretty quickly.
The only two usages of up in the original question that are standard are picking up the phone and getting up, where both describe a literal raising of something.
The other usages are colloquial or slang at best. To use them in writing would be sloppy (unless it was dialogue). To use them in speech would sound odd to people without the same accent.