There was a standard Chinese dialect during the Han Dynasty and Three Kingdoms. Yang Xiong's 1st-century Light-Carriage Messanger's Explanation of Other Countries' Local Expressions in Times Past distinguished the empire's 通語 ("common language") from the 方言 ("dialects") spoken by various regions within and around China. This standard was also known as 正音 ("correct pronunciation"), 雅言 ("elegant speech"), and later 天下通语 ("universal common language"). These strictly referred to the standard spoken language for the empire.
The prestige dialect during this time was 洛語 ("Luo"), the Chinese spoken around the Eastern Zhou and Eastern Han capital Luoyang. In English scholarship, it's usually discussed as Eastern Han Chinese or Old Chinese, although the latter can technically include other dialects going back as early as Sino-Tibetan. Still, when you see Baxter, Sagart, & al. reconstructing Old Chinese it's usually for the Chang'an (=Xi'an) and Luo dialects that developed into Middle Chinese.
The present-day Luoyang dialect (洛陽話) has different tones than Mandarin, which is based on the Beijing dialect which became standard during the Ming and Qing dynasties. It also still has the sound /v/ and some different vowels. Old Chinese was very different: its syllables were longer, with many consonant endings, and it probably lacked tones altogether. Nothing in China today is quite like it, but the southern parts of China were settled later and stayed poorer longer so they've preserved some things: 中國 in Mandarin is Zhōngguó but in Luo it was probably closer to Trungkwək or Tungkwug; you can still see traces of that in Cantonese Zung¹gwok³, Gan Zung¹guet⁶, Hakka Zung¹guêd⁵, Eastern Min Dṳ̆ngguók, Hokkien Tiong¹kok⁴, Shanghainese Tsonkoq. You see it in foreign loanwords, too, like Japanese Chūgoku, Korean Jungguk, and (almost perfectly preserved) Vietnamese Trungquốc. It also had a bunch of glottal stops, although they (and the final consonants) were starting to disappear by the Three Kingdoms period.
The written form of Old Chinese survived a lot longer so it gets a separate name: Ancient Chinese. Chinese students still have to learn it so they can appreciate almost anything written in the country until the early 20th century. It shows us that Old Chinese also used very different vocabulary and grammar from modern Chinese and (especially in writing) was very terse.
- The standard dialect of the Eastern Han dynasty was Luo, although it was called other names at the time.
- Neither Cao Cao nor Liu Bei grew up in the capital, so they'd use their local dialects informally with old friends. With people from other prefectures or in formal contexts, they would've used Luo.
- You can see the Baxter/Sagart and Zhengzhang reconstructions of Old Chinese pronunciations in most Chinese entries on Wiktionary. They're similar but not identical, because we can't be 100% sure what Old Chinese sounded like. Still, looking at phonetic character radicals, poetry, rhyming dictionaries, misspellings, Chinese dialects, foreign loanwords, &c., linguists can make some very educated guesses. It was something like Vietnamese or Cantonese without tones (not because it came from those areas but because those languages have preserved the older forms more, the same way Shakespearean English sounded more Scottish or Appalachian than like a present-day Londoner).
- The Luoyang dialect for this period would've been a little softer than those, with some or all of the hard final consonants disappearing.