As far as airline travel, it is not dangerous during pregnancy. Several concerns have been raised about whether there is an increased exposure to radiation on an airline flight. While this slight increase is measurable, it is not significant and does not increase risk of birth defects or other pregnancy problems. There is no evidence that low cabin pressure makes it more likely to go into labor or rupture membranes. That doesn't mean travel is dangerous, it can just make it a lot more exciting.
Morning sickness is a common occurrence during the first few months of pregnancy.
There are several ways to make the discomfort of morning sickness more bearable. Some women do well to have a few saltine crackers or pretzels at the bedside to eat before arising. Many women will want to avoid any activity that has to do with food. This would include grocery shopping, preparing and cooking meals, and washing dishes. Large meals are to be avoided.