My go to tactic is emphasizing the fact that most people already agree with being vegan. Most people actually have the same morals. I've never spoken to anyone who can disagree with the statement that "hurting animals for pleasure/habit/convenience is wrong". Read Eat like you care. It will help you a lot in these conversations.
There is a small minority out there who do truly think that animals are here for them (often based on religious claims) or who honestly aren't bothered by the idea of animals being hurt. Ignore them. Focus on the other 99% of people who do have some moral concern and moral impulse towards animals. Get them to admit that they agree with the basic premise and that their actions do not agree with their own morals.
I love the book Eat like you care. It's actually more like a 'Not eating animals for dummies' style of writing than an in depth philosophical treatise, but if you enjoy logic and reason, I highly recommend it. What I like about the book is that once you finish it, it helps you to mentally distill all of the million and one different reasons to go vegan down into one very simple principle. Personally I have a supreme intolerance for logical fallacy based conversation (and this will account for 99% of the conversations directed at you), and this book really helps you to quickly identify them when people bring them up so you can have more productive conversations. Even if you don't plan on being some vegan advocate or even telling anyone, trust me people will notice, and when they do they will by golly want to let you know all about what they think.
Please just remember, that you are not wrong. To go against the tide like this means that it is some fundamental truth that you have recognized from the very core of your being about such a simple thing - not wanting to hurt others. That cannot be wrong. And hopefully you and your mom can find some understanding, but also keep in mind that to go vegan, you will always encounter this resistance. If she comes around, there will always be extended family members, friends, etc so in the long term, you must be able to accept it to some degree as well. Best of luck, and come back for support anytime! I like to think of everyone I know as potential vegans. After all I was just like them not so long ago.
Also, I highly recommend this book to help when responding to others. It's very short, takes only an hour or two to read, but is an excellent resource for learning how to have productive conversations with people about veganism. Digital download is also available.
From the book description: "The authors explore the 30+ excuses they have heard as long-time vegans and address each one, showing why these excuses don’t work. Packed with clear, commonsense thinking on animal ethics, without jargon or complicated theory, this book will change the way you think about what you eat."
Eat Like You Care by Gary Francione. It's a lifechanging book for so many people.
>Is locally grown food as bad to you?
Environmentally, yes. It's not just the distance itself that the grains are transported, it is the massive difference in the amount of grains that are required (or the amount of grassland that is destroyed), the pesticides and fertilizers that are required, the water that is required, etc. And of course, ethically, yes. Cow doesn't feel better being shot in the head or getting her throat slit because the guy down the road is going to eat her.
Convenience foods in restaurants/grocery stores are definitely going to be harder to find than non-vegan ones. Perhaps you can give more details about "on the go", like does this mean OTR truck driver, or you just tend not to pack a lunch for your office job, or you're a college student, or...? If we know more we can probably help with specific restaurant/grocery store meals, or things that are easy to pack, or easy to make ahead in bulk/freeze, etc.
Re: acne, vegan diet may or may not help much. My advice is get a well formulated (alcohol/dye/fragrance/other irritant free; at proper pH) chemical exfoliant, benzoyl peroxide too if you want, wait a couple months, and your skin will most likely clear up.
A couple resources that honestly, will cost you about $20 altogether but I think are among the best and most informative out there, are this very short book for more of the really common questions like you ask in #1, and the documentary Cowspiracy for more on the environmental impact.
Looks like you got a jump on the basics but depending on how much you've researched the ethical topics below are some great resources to help keep you motivated. And once you understand the ethical necessity of a vegan lifestyle you'll find it is really a breeze!
earthlings - very graphic film but helpful to remove any doubts one might have regarding how we treat animals
eat like you care - a short and concise framing of the logic behind the ethical rational for veganism
Why We Love Dogs, Eat Pigs, and Wear Cows - another relatively easy read which lays out a bit broader argument
There are loads of other great books and films out there and I really suggest taking the time to educate yourself about the way we use animals as it will make the transition so much easier.
Good luck and welcome!