Malaysians normally communicate with each other in English, or more specifically, Malaysian-English, which sounds very close to Singapore-English, so-called because we sometimes add a local Chinese or Malay word here and there into our English sentences.
Malaysia (or, in its previous incarnation, Malaya) was under British colonial rule for 170 years (starting with Penang in 1786, Singapore in 1819, then the other Malay states), so English was widely used in business/commerce and everywhere in our daily lives ever since. Standard of English is maintained in Singapore, but has deteriorated in Malaysia in the past decades - but you’ll still meet street hawkers, cab drivers, etc. who speak English passably.
Education in Singapore uses English as a medium of instruction - it was the same in Malaysia until 1971 when Malay replaced English as a medium of instruction (Malaysians still learn English as a second language in school).
Malaysians of different races can also opt to communicated with each other in Malay, though that’s rarer in Singapore where English remains the main language.
Chinese of various dialect groups in Malaysia/Singapore (Hokkien, Cantonese, Hakka, Hainanese, Teochew, etc.) will communicate with each other in English, or else Mandarin. Indians of various ethnicities in Malaysia/Singapore (Tamil, Malayalam, Punjabi, Bengali, Telugu, etc.) will also communicate with each other in English.
Living in urban centres in Malaysia like Kuala Lumpur and Penang is akin to back in Singapore for me, where English is spoken widely - but if you move to smaller towns in Malaysia, you may have a problem if you don’t speak Malay. I usually switch between speaking English, Malay or the various Chinese dialects to people I meet without being aware of it, actually - but I always start with English as a default language at the beginning when I speak to public bus drivers or cab drivers.
Malaysia and Singapore are indeed ethnically diverse - Singapore has a 78% Chinese population (mainly Hokkiens, but with smaller numbers of Teochew, Hakka, Cantonese & other dialect groups), Malays form 14% of Singapore’s population, followed by Indians (7%) of mainly Tamil ancestry. Penang has about the same composition (also mainly Hokkien dominated), whereas Kuala Lumpur has a 45% Chinese populace (mainly Cantonese, like in Hong Kong), 43% Malay and 7% Indians, mainly Tamils.
The different races have lived together for 200 years, so we’re very familiar with each other’s culture, mannerisms and cuisines. Where else in the world would you be able to see Indian toddlers wield chopsticks with great dexterity to eat Chinese noodles, or Chinese 3-year-olds using their hands, Indian-style, to eat South Indian parathas with dhal curry for breakfast?