Enter the Soul Cairn. Nothing Happens. Speak to Serana. She tells you that the Soul Cairn is trying to take your life essence as payment. There are two options for going into the Soul Cairn:

Vampires aren't counted among the living. You would have to become a vampire to get in

Or,

The Soul Cairn wants a soul, so you will give it yours. Serana will partially soul trap you, and offer that gem to the Ideal Masters. It might be enough to satisfy them. It would make you a bit weaker when we travel through the Soul Cairn, but we might be able to fix that once we're inside.

Your objective is to:

Become a Vampire

Or,

Become Partially Soul Trapped

What's Better, to let Serana Soul Trap you or to have her make you a vampire?

Let's look at the two options:

What does "Soul Trapped" mean?
If you choose the soul trap option, it means that you get the following magic effect (debuff):

Weakened Soul
Health, Magicka, and Stamina recover slowly while in the Soul Cairn and are reduced by 45 points.

The penalty is not so significant, especially if you are high level. You should still be able to clear the Soul Cairn even with your soul being partially soul trapped. However, if you do run into difficulties killing the enemies inside the Soul Cairn, you could always turn down the difficulty through the menu.

Can you be a Vampire Lord and in the Dawnguard in the same time?
The Dawnguard will NOT be happy if you become a Vampire Lord. They will not help you, or give you side quests (though you can still finish the main story quest).

Vampire Lord or Soul Trap?
The best thing is to NOT become a Vampire Lord at this point in time, if playing as a Dawnguard. You will still be able to talk to Serana and have her make you a vampire lord later in the game, so it's best to wait until you have finished all the Dawnguard sidequests. So for now, you should opt to become Soul Trapped.

Once you've made your decision, talk to Serana again. You can tell her:
I don't see another way. I'll become a vampire
Soul trap me. I won't feel right as a vampire

If you tell Serana to Soul Trap you, she asks you if you are sure. You will remain mortal, but you'll find yourself weakened within the Soul Cairn.

If you tell Serana that you will become a vampire, she tells you that you'd become the very thing you've sworn to destroy. She does not know how the Dawnguard will react. Tell her you are ready. Serana says that turning someone is a very personal thing for vampires. It's intimate, for us. She does not want you to feel like she is forcing you into it. Tell her that you are glad she is the one doing it. Serana bites you, and you become a Vampire Lord.

Enter the Soul Cairn. This completes "Chasing Echoes", and starts "Beyond Death".