Let the contractor organise any necessary road closure with the highway authority and the police. Leave dead wood in logpiles for wildlife. Ask for a written quotation and specification. You may want to ask two or three companies for quotations, but the cheapest may not be the best value for money. Trees can be important for wildlife, especially squirrels, birds, insects, birds, bats, and lichens.
Trees give shelter and food.
Ancient or veteran trees are especially important. Some species and habitats are protected by law under the Wildlife and Countryside Act , and you, and the contractor, could be prosecuted for damaging them. For example, you should not tell a contractor to remove a tree or to do tree surgery which would destroy shelter for bats, red squirrels, dormice, or nesting birds. Woods, especially ancient woods, are important habitats. If your garden includes a wood or part of a wood, it will be better for wildlife if you manage it as a whole habitat - it is not just a collection of trees in an ordinary garden.