Strictly IMO: there's a reason most groups don't bother with language stuff, just like most groups don't bother with detailed weight allowances or precise ration tracking. It's more tedious than entertaining for most groups.
However, I do think it's possible to make it relevant and interesting, even in a game where you have "Common" and such. You can:
1. At the outset, tell your players that "Common" is simply the most widely-used language, not the universal language, and that other languages (racial/cultural, class, or otherwise) may be relevant sometimes. This preserves the "day to day interactions aren't impeded" principle, while providing opportunity for a situation where a lack of understanding creates meaningful effects. As long as the party has multiple, varied languages, different PCs can all get a chance to speak on behalf of the party--or interpret for those who can't pick up on what's happening.
2. Make sure that some of the language differences aren't actually "social roll" stuff. For instance, perhaps all the nobles of the Dwarf kingdom know how to speak Common because that's considered a necessary part of formal education, but all official court speech must occur in Old High Kazadian ("Dwarf"). Informal discussions aren't impeded, but having a Dwarf (or someone who knows the language) in the party makes for a major difference in tone...IF the DM handles it correctly! (See next point for details on that.) Although language is fantastically important for how individuals socialize, it's also super important for things like acquiring information (in 4e terms, "Streetwise"; in 5e terms, kinda-sorta "Investigation") or concealment (can't "blend in and disappear" if you don't know all the little niceties of pushing your way through a crowd!). There's also all the possible sources of written language, as well as spoken, which can make a big difference.
3. STOP handling all "hidden" speech by sharing it with the whole party, as long as one person understands. To make it actually FEEL like a foreign language, the intelligible part should come exclusively out of the mouth of the person who does know it. (If that player has IRL speech difficulties, alternate measures are of course perfectly cromulent, but the preference should be on a level of "interpretation.") Extra-special points if you attempt to sound out a garbled gibberish sentence (or a gibberish sentence in a real language nobody else at the table speaks, e.g. German for players that only speaks English, Spanish, and Japanese).
4. Come up with a handful of important concepts, metaphors, and set-phrases unique to each non-Common language possessed by the party--and possibly even a few for Common as well. Tell only those players of characters who know those languages these special terms, and ask that they refrain from sharing them with the party until they actually come up in play. This will heighten the feeling that these are actually different languages, and not merely "the language all us players are speaking, dressed up funny." I'm speaking of terms like "schadenfreude," which doesn't translate well to plain English* but is an important and commonly-understood concept in German; of idiomatic phrases like the Spanish "tomar el pelo," which literally translates as "to take the hair" but is really a metaphor, roughly meaning "to mock in a good-natured way"; and (say) how American English uses "How are you?" as a greeting and "You alright?" as a question of concern, while British English does exactly the opposite (according to Wikipedia anyway). Little linguistic flourishes like this make a huge difference in communicating a feel of other-ness, while adding very little actual labor to communication.
5. Give players the opportunity to actively improve their language skills. Initially, their knowledge is zero--they simply straight-up fail any check that would absolutely require knowing the language. However, if they make a concerted effort to learn (e.g. spend significant one-on-one time with a native/fluent speaker, such as a party member who knows the language), gradually step things up. After, say, 5-10 sessions of concerted effort, things start to pay off. You don't understand the language, but you can roll a flat check (no bonuses) with disadvantage, to attempt to puzzle out what was said. If you're within your Proficiency bonus of the DC, you figure out some but not everything (and you know this). Beat it and you have the gist even if you miss a few details; fail at less than proficiency and you either didn't understand at all, or you think you understood but are wrong (might be best to have the DM roll this secretly). Let it advance further from there--another 5-10 sessions and you can add your proficiency to the check, another 5-10 sessions after that and you no longer fail to understand completely, all fails count as the "within your proficiency value" effect. And finally, after a grand total of 20-40 sessions (or whatever scale you prefer) actively trying to learn the language (an extremely long time for most characters), you understand well enough that you no longer need to roll unless the situation is clearly bad for communicating, e.g. high winds, listening through a door, etc. You still cannot reliably speak the language yourself, and require a proper teacher to learn how to read and write it, but you can now at least get a basic understanding when the language is used in your presence.
6. Consider giving extra languages at character generation for high Int score. Currently, Int does very little other than a handful of skills and Wizard spells (and those derived from it, that is, AT and EK). Perhaps even subtract a language for a low Int mod: you only know either the Common tongue or the language common to your people (race/culture) rather than both. Gives a carrot to those who invest in good Int, and a minor but relevant stick for those who dump it.
*The dictionary definition, "pleasure derived from the misfortune of others," is iffy. The term can also apply to the relief felt when misfortune falls on another instead of you--no real desire that the person be hurt, but still being happy because somebody else suffered. It's also seen as distinct from sadism because the latter generally entails a desire to (personally) inflict pain/suffering/misfortune, while schadenfreude definitely doesn't require that.
