Possibly, and most likely, you don’t understand yet that the object is placed before the verb in French in all cases when the verb is a pronominal verb and that’s what “se connecter” is; in English, you say “connect yourself”, in French, “vous connecter”. It follows that “vous” is not the subject of “connecter” but its object.

• Begin to connect yourself with your friends today.

This is true only in this type of construction (infinitive) and when the verb is conjugated;

commencer à, finir de, continuer à, …

• Commencez à vous renseigner. — Finis de te laver. — Continuez à vous préparer.
• Vous vous renseignez. — Tu te laves. — Vous vous préparez.

When the verb is in the imperative, the inverse is true;

• Renseignez-vous. — Lavez-vous. — Préparez-vous.

My gloss for this would be:

Start connecting with your friends today

An overly literal gloss would be:

Start to connect yourself with your friends [from] today

The commencer is in the imperative, and connecter is prefixed with vous because the verb is reflexive here. You are connecting yourself to your friends. One might think of it as the necessity of having both something to connect from and something to connect to; there must be (implicitly) a plurality of things that are acted upon.

It may be helpful to rephrase this as a declarative statement, so that the subject of the sentence (and thus, reflexive nature of connecter) is more obvious.

Vous commencez à vous connecter avec vos amis dès aujourd’hui.

When meaning "to establish a connection between oneself and something/someone else", to connect is intransitive in English whereas se connecter is pronominal in French so your sentence translates to:

Start to connect with your friends right now.

Its word for word translation back to French would not make sense:

Connectez avec vos amis dès maintenant.

Se connecter can also be considered a reciprocal verb here:

You and your friends, start to connect with each other right now.

Vous et vos amis, commencez à vous connecter ensemble dès maintenant.