WASHINGTON—President Donald Trump starts his first week with a packed schedule—from pushing through his slate of cabinet nominees to a raft of executive orders and setting the direction on foreign trade.

Mr. Trump is planning executive actions early in the week on immigration and trade, two White House officials said, and will have a chance to lay the groundwork for a trade deal during a meeting Friday with British Prime Minister Theresa May, the first foreign leader to visit the new president in the White House. He also will...