The correct form is: Io ci tengo a te. This sentence almost always has also 'Io' in it, reason is that it is a strong assertion about feelings. Using the shortend version 'Io tengo a te' might actually sound 'hard to believe' as it has not the due emphasis, so it could even reach an opposite result from what was intended. Another form, used even more often: Io ci tengo molto a te.

Both possible, it depends on the context, if you use tengo without "ci" you must specify what you care about. The most used is:" ci tengo a te" "ci" has many functions, in this case the function is to reinforce the expression. some more examples: Non capisco, tu tieni più a me o al lavoro? Certo che tengo a te, ma tengo anche al lavoro. oppure Certo che ci tengo, ma tengo anche al lavoro.

Tenerci is a 'verbo pronominale' that means 'I care about something', usually goes with the indirect object (a qualcosa) and almost unavoidable 'CI'. Ci tengo molto al mio sassofono, alla tua salute, alla tua formazione, al nostro rapporto. I wouldn't use it with a person, I wouldn't say neither 'Tengo a te', nor 'Ci tengo a te', nor 'Ci tengo alla mia professoressa di italiano' but 'Ci tengo alla nostra relazione' or 'al tuo benessere' or 'alla nostra famiglia'. Tenere without CI has another meaning, it's something similar to 'give' and 'give up'. CI in this case is both an indicator of the phrasal verb and a repetition of what you care about. (ci = al mio sassofono, etc.)