Is it normal I think my skin looks worse after my first chemical peel?

Chemical peels can be a good option for hyperpigmentation.  And you are correct that sometimes it takes a series of peels to achieve the desired result.  The type and strength of peel help determine how significant of a result you will get, but there are many other variables to consider.  I would think that you would see some type of result after your peel.  It may be beneficial, both medically and financially, to schedule a consult with a Facial Plastic Surgeon to get another perspective on the issue.

If you are new to aesthetics, chemical peels are a great way to start sloughing off the dead layers of skin to reveal a more youthful look.  Some patients experience a slight purge after a chemical peel or any aesthetic procedure for that matter.  Your body uses the skin to push toxins out, which sometimes can result in the skin looking worse before it gets better.  If you experience a purge, do not worry because it is normal.  Talk with your provider and make sure you are on medical-grade skin care to start treating your skin.  You will love your results as you continue your treatments and correct products!

Pimples after a chemical peel could happen because your skin is disrupted by the chemical peel. Your skin is trying to adjust with the treatment. Chemical peels do help lighten the pigment with a series of sessions.  It is a skin condition that is very stubborn to treat, I would have a consultation with a dermatologist to evaluate your skin first to see if you need any prescription topicals to make your treatments effective. 

I don't know what kind of peel you received.  Most of the time, peels will not make your skin look worse.  They may not improve the problem after just one peel. What is the home care you are using?  I usually recommend patients use a fading cream/glycolic product for a couple of weeks prior to a chemical peel if they are trying to get rid of melasma or hyper-pigmenation.  The pre-treatment regime helps augment the effect of a peel.  Are you using sunscreen?  Have you been out in the sun?  Sometimes you can get acne after a peel, but this usually goes away as the peel comes off.  I would ask your esthetician what she can suggest to use at home.