I want you to know that I created an account on this forum just to answer your question. I found your post when I was searching for a game's quest list. Most of the people who have answered you have been very dismissive of what you're going through, and I wanted to make sure that at least one person answered you with genuine concern and respect.
It is hard to give helpful advice to a person without really knowing them, so please understand that most of this commentary is general and also based on my own experiences which aren't exactly like yours.
I am a geek. I am super smart, I am a chick, I play geeky games, and I have a group of amazing friends who share the same qualities. We are successful people with well-rounded lives who - *gasp* - play card games, video games, board games, and roleplaying games. None of us play Yugioh or Pokemon, but that's 'cause we're not part of your generation. We also go out, party, hang-out, go to concerts, the movies, football games, and have families. My friends range from computer programmers, professors, teachers, physicists, recording artists, lawyers, chefs, volunteers, political activists, and parents. Having fun playing games isn't stupid, and it doesn't have to stop when you're an adult.
People in life are going to pick on you. This doesn't stop when you get older, but it does diminish. There are ignorant people and intolerant people and hurtful people and petty people everywhere. This is much much more common when you're younger, and it's also harder to get away from. You have a limited life as a kid. You don't have money and you don't have the legal right to make your own decisions. However, this changes dramatically when you get out of high school and especially when you graduate college. When you're older, you gain control over what you do, where you live, and who you spend time with - this gives you the power to cut out the people who are toxic and keep the people who truly appreciate you.
Life will get better LATER, but you do have some power to make it better NOW, too. There are different kinds of picking or teasing. Teasing can be playful and loving. Many families and friend circles adopt a sarcastic and picking tone because that's how they are. However, it should be OBVIOUS that it's a joke meant to be shared WITH you, NOT at your expense. They should also provide encouragement and reminders that it is ONLY teasing. However, if you feel that all of these jokes are simply to belittle you, to make you feel bad about yourself, are coercive to make you change your actions or who you are, or to make the person picking on you feel good about themselves, then this is BULLYING, and this is NOT acceptable.
I think your dad is teasing you. I think your brother is bullying you. Let's start with your dad.
Your dad is *probably* teasing you because he doesn't relate to you well. You sound like you only have visitation weekends with him because of a family divorce (this is an assumption, on my part; forgive me if I'm wrong), so that means that he only gets to see you occasionally and thus doesn't have as much time to bond with you and get to know you for you are. This is going to get even harder as you get older and start having a life of your own and have even less time for him. The reverse of that is also true - you don't know HIM as well as you should either. He loves you but he feels awkward because you are sort of a stranger to him and you are getting more and more different as you grow up AND you are avoiding spending time with him on the few weekends you get to visit.
As a kid, I spent many awkward weekends with my distant, divorced father. On weekends he was feeling good about himself, we shared some pretty awesome hobbies that he introduced me to: archery, gun marksmanship, and photography (now that I think about it, they were all "first person shooters", ha!). But on bad weekends, I would spend most of my time reading books, playing games, writing, and watching TV so I didn't have to talk to him. I eventually stopped visiting him altogether. Now that I'm an adult, I see him once or twice a year and have very fundamental disagreements about life, politics, and how to treat people. I don't know if I will ever have a good relationship with him, but I am thankful to have at least some connection to him. I also know that if we hadn't had that bonding time when I was a kid, he'd be a complete stranger to me.
You need to have a talk with your dad - one on one, NOT with your brother around - and you need to explain to him that his teasing you and his allowing your brother to aggressively bully you is hurting you and making you feel like he doesn't love you. It makes you dislike spending those weekends with him. You need to say this respectfully and you MUST have the attitude that you want things to change for the better. This requires that you be open and willing to compromise on things, like sharing a hobby that he enjoys, too. Don't just expect him to do all the changing. Write up a little checklist of what you want to say before you sit down to talk to him. This'll help convince him that you're serious. Remind him that you love him, that you want to make these weekends enjoyable for the both of you, that you want to get to know him as a person and you want him to get to know you for the person you really are.
Invite your dad to play Yugioh (build him a competitive deck so he has a chance), and explain to him that it's not unlike other games he's more familiar with like "You Sank My Battleship" (attacking and subterfuge), Fantasy Football (selecting players or fighters based on their abilities, and building a team of them), and Risk. The tools and mechanisms are different, but the concepts are not alien. If he doesn't enjoy playing your games, tell him that you're willing to alternate activities, i.e. for every hour he spends playing something you like with you, you'll go to the batting cages or the zoo or the movies or help cook dinner with him. Remember, this is about making a lasting friendship and relationship with your father that makes you BOTH happy.
Now, about your brother. It sounds like there's a power struggle going on: for attention, for control, and sadly, for his own amusement. I'm certain there's probably some picking coming from you, too (Do you pick on him, too? Be honest!). It's possible that your brother is being bullied at school and takes it out on you when he gets home because he can't defend himself from his own attackers and you're an easy, younger target. Or it could be that he's just your typical kid who is self-absorbed, doesn't think about how his actions affect others, and doesn't get that the world isn't all about him. MOST people grow out of that. Many people who have bad relationships with their siblings when they're young wind up having great and strong friendships with them once they don't have to live in the same house together and see each other every single day.
I think you should do three things. 1) Try to get to know your brother. Don't try to talk to him the exact same way you have that sit-down conversation with your dad. Your brother just won't get it. He is probably trying to hurt you because he's hurting himself and doesn't understand how to make it stop. And as a good brother, you can try to help him feel good about himself. That doesn't mean that you should point out his problems and discuss them in detail. You should instead take little actions to give him some legitimate confidence. Invite him to go play basketball or ask him to teach you how to do something that he knows that you don't, i.e. how to change a tire, help on your homework, or anything that he likes to do, even if you don't think it's cool. Compliment him every once and a while (genuinely, not sarcastically), like, "cool shirt, bro" or "I like your hat"; it'll put him at ease and not feel like he's got to hurt you to feel good about himself. I know it's hard to even imagine doing something nice for him when he's been such a d-bag to you, but all you can do is try to be a good person, the person you WANT to be, and if he can't respect that, that's his problem.
2) Protect yourself. Keep your computer, your files, your social networking accounts, and your email password protected. Change the passwords regularly. I'm sure you already do a bunch of this. If your brother's lying and bullying start to affect your friendships with other people or your school, you need to talk to someone who can help you: your mom, your dad, the school counselor, anyone who cares about you and has the power to protect you. You brother may not be hitting you, but emotional abuse is real, and if he crosses the line (which it sounds like he already is), then you need to tell someone NOW.
3) This one is about you. You've used the word "hate" a lot in your post and you've talked about how you want to punch and kick your dad and your brother. This is rage, and the reason you feel this way is because you feel powerless to help yourself. You're hurting, you want the hurting to stop, and you want to hurt them back so they know how it feels. Sound familiar? It should, because it's what your brother is doing to you, only verbally. Don't become a bully yourself. Physically harming people is NOT OKAY, just like how your brother making up lies about you is not okay. When you start to imagine hurting your brother or father, go for a walk or run. Lift weights, channel that aggression into something else. This kind of anger can be very creative if you let it - just use it for making yourself stronger, NOT becoming a bully yourself.
I'll keep checking on this post to see if you have any questions. I didn't talk much about my life, but if you want to know where some of this advice is coming from, just ask, and I'll answer your questions as well as I can.
Remember, you have real friends out there, but your best asset should be YOU.
I really hope this helps.
Edit: grammar, explanation of an assumption