[action verb + 起来] does indicate beginning of an action, e.g. 动起来 = start to move; 打起来 = start to fight; 認真起来 = get serious

[action verb + 上] doesn't indicate beginning of an action, it indicates 'action toward an object' e.g. 爱上他 = fall in love with him; 算上他 = counting him; 走上絕路 = walk toward an dead end