You should insist on a signed contract. Requiring work be completed according to ANSI A300 Standards is advisable. The American National Standards Institute can help you better define how work on your trees should progress. They define techniques for pruning, fertilization, safety, and other categories of tree care. Your arborist should have a copy.
Cost is a factor.
Hiring to cover a one-time takedown with no repeat work in the middle of an old field, is one thing. But if your house and yard are in jeopardy, hire people who have the knowledge, experience, and equipment to protect your property. Many reputable companies can invoice their work in stages or on completion and receive payment on receipt of the invoice or within two weeks. A high price doesn't guarantee quality, but a good established arborist won't come cheap. Ask for recommendations, but consider the credibility of your sources.