I don't have all the answers but I can defenetely share my experience since you seem to be having the same problem that I had in the first trimester so...
EATING: I was throwing up at least 1- 2 times a day no matter what I ate. I figured after a while that I had most problem eating warm food, because it was releasing smell, so naturally switching to cold food seem working better. Especially sandwiches, dry bread of all sorts and danish alike and pasta things... but I figured that even throwing up so much I was gaining waight like crazy. So my next idea was to just cut on the amount. I still had to have starchy food but than I tried to have only like third of what I would like to have to cut on gaining waight.
Also, be carefull what they write about eating for two... I noticed that my metabolism not only slowe down like crazy - so technically I needed even much less food than before pregnancy to keep the waight or even gain not to mention eating more !!! higher calorie intake is right but for much later in pregnancy but that you will know yourself when you will be eating and not gaining too much.
Also... I figured that I had problem eating veggies, salads and fruits so I wanted to keep up with vitamines. So it was easier for me to dring those Naked or Odvalla ready made fresh juices than eat fruits.
Citrus -big problem! I felt sicker after orange than anything else.
EXCERSIZE - walking is almost always good idea. I had my first doctor visit late - not in first trimester so I had to play safe as I was not sure aobut excersize either. Technically there is many things that you can do to harm baby and without knowing what is good it is better to hold. as resting do as much good as excersizing. So I was walking and resting. And than when the doc saw me I asked if it is safe in my particular situation to excersize. It was okay. Asking doctor is good idea but still I am a bit conservative to what I do as it is after all my baby and I don't want to harm it by overdooing anything.
As is with swimming.. technically safe, but... even when I was pick of health I sometime got some rashes or infections by swimming in the pool with chemicals.. You never know who is swimming before you and how good desinfectants are there. and after all this is water and water can get everywherhe
feel good, stay healthy and good luck!