As someone who is not very empathetic by nature, I found Authentic Relating practice (check out, for example, www.authrev.org) very helpful for cultivating empathy, as it literally focuses on and trains "getting someone else's world." It also trains awareness of and ability to share your own emotional and somatic experience, which is central to emotional intelligence more broadly. I liked it because it was fun - it felt very connecting (I would leave events with a feeling similar to having cuddled with people, even when no cuddling had taken place - oxytocin something something), and I found exploring my and other's experience to be a rich and interesting experience.
Nonviolent Communication (NVC) (check out the book by Marshall Rosenberg, and also the online pay what you want Communication Dojo classes by Newt Bailey) is also very helpful for both empathy and emotional intelligence, as it systematically cultivates a empathy for your own and others' feelings and the needs behind those feelings. It was less rewarding as a recreational activity than Authentic Relating, but Newt is hilarious and lovely so his classes are really fun (and you can attend them on a one-off basis without committing to a series).
Both those practices have had a major impact on my ability to navigate relationship challenges (romantic and familial) with less anger and irritation and more success, as well as my own well-being outside of a relational context.
Lastly, I've read in passing (https://docs.google.com/document/d/1nrHi6vTRJI_MELW_gtTiEaaYwK8I82ytMIpHbmM0KNY/edit?usp=drivesdk) that metta (loving kindness) meditation improves empathy. In my experience, it does cultivate a feeling of wa friendliness and care towards other people, but is less helpful in providing insight, compared to the other practices I mentioned. My experience with it is more limited, though. A neat bonus is that once you get some experience with it, it's like a "muscle" you can use in situations where you might otherwise feel anxious or irritable - e.g. I've silently wished loving kindness while going through an airport, listening to a crying baby on a plane, in an annoying or frustrating meeting at work, and before parties where I didn't know too many people, and it really improved the way I felt. It can also feel more immediately rewarding than some other types of meditation.
All these sound super hippie, I grant you, and you may have to hold your nose initially if you're allergic to that sort of thing (I did), but they're well worth it.
