TL;DR: Massage is one of the best methods of managing pain, stress, and mood disruptions brought on by pregnancy. Where it does not help, physical therapy likely can. Exercise, both aerobic and strength training are important for mother and her growing fetus.
Massage from a skilled provider is not dangerous; it is helpful and perfectly safe for most pregnancies. Exercise is not dangerous; it is recommended for most pregnancies. If you have concerns, finding an educated provider is key.
Prenatal massage is a topic I am passionate about, and have written about on my blog in the past. Many issues emerge from the poor education standards of massage therapists, misperceptions about the fragility of women, and a multitude of myths perpetuated within "alternative medicine" circles. As if pregnant women do not have enough real concerns to keep in mind, false anecdotes are spread about exercise being unsafe, massage causing miscarriages, and reflex points* that induce premature birth. Most of these concerns come from a "better safe than sorry" mindset, but the actual risks associated are overblown out of ignorance.
First, to say it absolutely unequivically, massage in the first trimester is not a risk to the mother or her fetus. The textbooks used to educate massage therapists are not usually written by experts on the subject matter and thus perpetuate many myths generation after generation. Social media is far worse, as having an audience does not correlate strongly with having knowledge on a particular subject. Massage is associated with multiple benefits including: reduced back pain, improved depression and anxiety, reduced stress, improved sleep quality, and improved fetal growth and development. Regarding inducing labor, there is absolutely no "Eject Button" that causes early labor on the inside of the ankle or anywhere else, as far as I can tell someone just made this up and attributed it to "merideans" without evidence.
Most massage therapists will put prenatal women in the side-lying position because that is what they were taught to do. This is a good practice because it is often the most comfortable position anyway. It is rarely medically necessary. The supine position (flat on your back) can be dangerous for women past the 20th week of pregnancy for some women. The actual percentage of women with this condition, called Aortocaval compression syndrome, is very small. With adequate use of cusions and bolsters, most positions can be made safe. For my prenatal patients, I use a set of specialized cushions which can allow for prone, supine with a left tilt, or side-lying. Although generally side-lying is my position of choice.
On the topic of exercise and physical therapy, expecting mothers are often given exactly the wrong advise out of misplaced concern. Exercise is not only generally safe for women throughout their pregnancy, it has been shown to improve weight management, reduce gestational diabetes, decrease incidence of preeclampsia, improve mood and self-esteem, reduce back pain, make labor go more smoothly, and improve fetal development. By all means, get out there and exercise unless you have underlying health problems or previously difficult pregnancies which place you in the "high risk" category.
If you have an abnormal pregnancy, or some underlying health condition, you are not alone. While you may have to follow some additional precautions, and see your doctor a little more frequently, massage and exercise can still be tailored for your unique needs. If you are struggling to find a physical therapist or massage therapist in your area, there are many resources available to help you find a qualified one.
*Reflexology is not "an effective treatment for any medical condition"