If you have a doctor who understands how dangerous vaccinations can be, then they should be able to advise you on which ones to start with. Some doctors can offer separated measles, mumps and rubella vaccinations, which is preferable to the triple-shot that seems to have caused so many problems for so many, even though it means more shots and trips to the doctor overall.
And yet... MMR may not be high priority, as measles used to be a run-of-the-mill, no-big-deal disease that would give you natural and uncontaminated immunity after it passed. I just watched a Brady Bunch episode where the whole brood came down with it in one day, and the most distressing part of the show was that Mike and Carol had called in two different doctors. Measles and mumps are not generally life-threatening, so not a high priority.
I would stick to the most debilitating and deadly diseases: polio & rubella. Polio vaccinations are not supposed to use live virus anymore, so do your homework. One of the most common reactions to the chicken pox vaccine is shingles. I'd rather have chicken pox than shingles! Whooping cough is scary but treatable, survivable, and recovery will provide natural immunity. HIB is like kennel-cough for toddlers, so I wouldn't bother with that, if your kids are at home.
And whatever you do... (oops, preaching!) stay far, far away from all of the Hepatitis vaccinations. When your 3-year-old starts having unprotected sex or injecting drugs, go get it. In the meantime, you don't need it. That was crass, but for Heaven's sake, they're giving them to NEWBORNS! Ridiculous...