[verb] + 起来 + 了
[adjective] + 起来 + 了
[verb] + 起 + [object] + 来
起来 after a verb says the action has started and is now going on. It usually comes with 了, and often with 突然 for something that started without warning.
After an adjective, 起来 says a state is setting in and building: hotter, warmer, busier.
When the verb has an object, it goes between 起 and 来.
开始 "begin" names the start as a plain fact and can look ahead. 起来 reports a start that has just happened, often a sudden one, and doesn't go with plans.
起来 keeps its literal sense in 站起来 "stand up", and has other uses – 看起来 "looks", 想起来 "remember" – that belong to a separate point.