That would be a variation of your second guesses:

Je prends contact avec quelqu’un.

Je reprends contact avec quelqu’un.

Your first sentence "j’entre en relation avec quelqu’un" is possible too but less common. It also implies a closer / longer relationship with the other person.

Rentrer en relation doesn’t convey the back (or again) meaning.

If you mean that you were in contact with that person and somehow along the way, you grew apart and now you are reestablishing your connection, contact … you can say: Je rétablis mon contact (ma relation, mon lien) avec quelqu’un..

As for prendre or reprendre, your formulation is wrong, on prend (reprond) quelque chose, je prends un café, le train, …, do not use prendre en in this case, prendre en is used in cases such as je prends en compte meaning considering or prendre en otage, abducting.

A third alternative (reprendre contact et rétablir le contact are correct, and it is correct that you mustn’t use the "en" in your sentence with (re)prendre ) is

renouer le contact avec quelqu'un

Like in the case of "rétablir", this implies that the contact has been lost (while with reprendre it is only after a short interruption). The difference is that "rétablir le contact" usually means that the contact has been abruptly broken (e.g. after an argument , when you need to "heal" the relationship) while "renouer" is more for old friends that you haven’t seen for a long time (it is still a friend, but he has been abroad for a long time so you did not talk to him anymore until he came back because the relationship was neglected). Those are just small nuances…