"I miss you" is one of those phrases that is difficult to express in Japanese.
Don't get me wrong — they do have similar phrases, and Japanese speakers are completely capable of expressing the concept of missing someone.
But something about the alternatives to the classic English "I miss you" just never felt all that satisfying to me.
That said, let's take a look at what we have to work with...
会いたい (aitai) = I miss you...?
会いたい (aitai) is probably the most common way that "I miss you" gets translated into Japanese.
Literally it means "I want to see you." Well, perhaps even more literally it means, "I want to meet with you," but I think that we can agree that those have substantially different connotations in English.
So, if you were going to text your lover "I miss you," but you want to say it in Japanese, then 会いたい (aitai) is probably the phrase for you.
There is also a Korean drama that, while translated in English as "Missing You," becomes 『会いたい』(aitai) in Japanese. Here are the English and Japanese Wikipedia pages to prove it!
恋しいこいしい = I miss you...?
First thing to get out of the way with 恋しい (koishii) is that you can only say it if it's impossible to meet with the person you miss (i.e. the person you're 恋しい [koishii] for). So you can't exactly yell it across the house to your lover.
This makes it slightly different than 会いたい (aitai), which does not necessarily mean that you are incapable for meeting with [missed person].
Another thing that sets 恋しい (koishii) apart is that it's totally okay to say it about a place or an object.
amerika no piza ga koishii na~
I miss American pizza!
If, on the other hand, you said:
amerika no piza ni aitai na~
I miss [want to meet with] American pizza.
Then it sounds like American pizza is a person that you're missing, which seems unlikely.
寂しい (sabishii / samishii) = I'm lonely (because I miss you)...?
Technically, 寂しい (sabishii / samishii) means "I'm lonely." However, Japanese people say it a lot to one another when they mean to say "I'm lonely (because I'm not with you." In a way, this makes it a kind of substitute for "I miss you," also.
Side note: either pronunciation of this word is fine: 寂しい = さびしい / さみしい = sabishii / samishii .
寂しい (sabishii / samishii) has automatic "I miss you" popups in LINE:
This just validates our assumption that a lot of people are typing 寂しい (sabishii / samishii), "I'm lonely," when really they mean "I miss you," or "I'm lonely (because you're not here)."
淋しい = Seriously, I'm lonely (because I miss you)...?
I asked Rei the difference between the two different kanji that are used to write this word: 寂 and 淋.
We talked about it for a minute and decided that the second kanji, 淋, has a graver connotation. As in, "I'm alone, and it's not OK." If you're just being a whiny lover, then I'd use 寂 in your "I'm lonely (because you're not here)" message.
If you want to pick up enough Japanese to actually get a Japanese speaker to talk with and miss, then NativShark is the place for you.
Good luck with your Japanese studies!