◆「ともだちにあう。」
"I (will) meet a friend."
The particle に serves as an object particle, as in the example here, where "ともだち" is the object of the verb "あう." Essentially, に indicates the person or thing an action is directed. So now, if it is, you probably think why you can't say, "ともだちを あう。", don't you?
Here is another example:
「ともだちに Eメールを おくる。」
"I (will) send a friend an e-mail."
With the verb like "Eメールを おくる。[I (will) send an e-mail.]", に indicates the receiver of the action. In English, the receiver corresponds to the indirect object.
To sum up, I could literally translate those sentences to English "I (will) meet to a friend." and "I (will) send an e-mail to a friend."
◆「いっしょに うたう。」
"I (will) sing together."
The particle に of the word "いっしょに" indicates the verb (also behavior) "うたう" for showing how it is done. This usage of the word "いっしょに" is the same as "ゆっくり(slowly)" like "ゆっくり うたう。[I (will) sing slowly.]", "はやく(quickly)" like "はやく うたう。(I [will] sing quickly.)" and so on. I think this に is, so to speak, "an adverbial particle" in this case.
I'm sorry for my English, but do you understand? Does it make sense to you?
If you have more questions, feel free to ask me.
Thanks.