While there are many options to choose from, most new drivers should expect to start in some type of over-the-road position.
Many companies also operate more regional routes, which could get you home more often, and some will even hire new drivers for their local routes.
We would recommend that new drivers try to start in an over-the-road position to gain experience while lessening their crash risk for at least the first year.
Most new truck drivers will start driving over-the-road. Most of the major carriers that hire new drivers do mainly long-haul, and it's the best way for drivers to gain experience while minimizing the risk.
The general rule to expect is 1 day home for every 7 days out on the road, though it will vary wildly from company to company and depend on the flow of freight.
Regional and dedicated jobs will normally get drivers home more often, from a couple times a week to bi-weekly, while local and pickup & delivery routes will be day jobs with drivers going home at night.
It is possible, and it does happen, but not very often. Most companies are going to want to see some OTR experience before letting them drive local routes.
Additionally, putting inexperienced drivers into complicated & crowded driving situations with frequent stops and deliveries increases the chances for an accident or incident, and we would generally advise getting some over-the-road experience before attempting a local gig.
Any control that you have in that regard will be in the company you choose, depending on what areas they drive in. Otherwise, you're going to be basically taking what they give you, to wherever they need it to go. Some companies will fire drivers for refusing loads, some won't, but it is still highly inadvisable to refuse loads.
There are many companies, however, that will not force drivers to pick-up or deliver in New York City, as it is generally accepted that driving in NYC is a complete nightmare.