You may always struggle with this is issue, whether it presents within friendships or outside of them.
Maintaining friendships with those who operate under substantially different ethical frameworks is important. We have a far better chance of persuading those with whom we enjoy rapport than with those we have alienated.
It can be worth asking yourself, "Have I always been a vegan?". If the answer is no, it is an acknowledgement that you were once in a similar state of disregard for animal welfare. This acknowledgement renders us less prone to arrogance when trying to convince others of our views.
You might draw comfort from the fact that veganism and vegetarianism are slowly but surely permeating the mainstream, to the extent that vegan and vegetarian products (including synthetic meats) are enjoying an ongoing upsurge in popularity. More and more people are making an ethically-based decision to reduce/cease their consumption of animal products.
I now offer some claims that might be deemed proselytisation, but I make them here to tackle some of the comments already made and to provide some facts which should be considered within philosophical contemplation of the veganism; including the notion of how you might engage a friend with the topic.
We know now that - in many developed countries at least - it is quite possible to maintain a healthy vegetarian/vegan diet at a cost and convenience roughly equivalent to that required by a normal diet (If to no-one else, I've proven this to my own satisfaction. For all my many faults, I'm a perfectly healthy, active, athletic, 46 year-old male who has maintained a vegan diet for approximately 5 years. My skin isn't yet translucent, I weigh 105 kilos, walk 50km per week and can still perform 10 ugly chin-ups. I have never suffered from any symptoms of B12 or protein deficiency).
There is still a perception amongst most people that veganism is something akin to extremism. There are at least a couple of ways to react to this. Of course, to cultures that have grown and flourished thanks to the exploitation of other animals for so long, it has been normalised to a point at which any suggestion that we should change our behaviour certainly seems silly or fringe or extreme. There is no way any shift in such entrenched dietary and corporate behaviour will ever be anything but snail-paced, even in the most privileged of societies.
Then there is the view that asks us to consider how basic concern for the fact animals are unnecessarily being tortured and killed might ever be deemed extreme. This view is powerfully articulated by a single question, crudely phrased as:
"What trait(s) do other animals possess or lack which justifies our killing of them?".
This is known as the "Name the trait" argument. This simple form is prone to some refutations, but is a fairly strong 'pub debate' conceptualisation of the issue. The argument is articulated far more rigorously here at Philosophical Vegan (The link is clearly to a biased site, but it includes refutations of the argument and provides a good basis for further discussion).
Whilst it contains some assumptions, 'Name the Trait' is essentially asking how the torture and killing of other animals might be justified given a consistent application of the views which govern how we treat our own species. It is this argument which might be useful in answering your final question: "Where do I even begin to define what he values as morally problematic or not?"
I have never come across (in personal debate) any sincere anti-vegan arguments which boil down to anything, ultimately, much greater than "because I like the taste of meat and I want to keep eating it", or "The industry is massive and employs a lot of people". Maybe the deep thinkers on this forum will pose some interesting answers to 'name the trait', which I look forward to, but if we survive long enough as a species, current trends suggest we will one day look back upon our treatment of other animals as a truly horrific age that took us an unfortunately long time to transcend, much as we now look back upon our treatment of indigenous peoples, women, ethnic minorities, children and homosexuality with an understanding that we have done well to change our ways.
I am not a psychologist, but I suspect much of our apathy towards animals comes from a phenomenon known as 'cognitive ease'; roughly stated, this is the notion that humans are drawn towards those ideas which cause us the least amount of psychological stress. To actually stop and contemplate the horror to which we subject billions of animals daily is, for many of us, to peer at an aspect of ourselves which is entirely at odds with how we like to think we are; at odds with the logic we typically apply to ethical issues. It is far easier to look away; to pretend there's nothing to think about.
For an interesting interview between the vegan and philosophy enthusiast Alex O'Connor, and the two skeptical hosts of Triggernometry, click here.