Turbulent relationships between stepchildren and stepparents are so common, it's a cliche. The evil stepparent is the theme of countless books and movies. Adapting to someone new in the house is hard for everyone, but you can overcome these challenges and become a happily functioning family. Try to see things from the kids' perspective, but don't let them steamroll you.
Present a unified front. It doesn't matter what you think about your partner's ex; your stepkids will manipulate you less if they know everyone's on the same page. Talk with the ex about parenting expectations and try to support her as much as possible. You'll cut down on the "My mom lets me do this at her house" comments. Don't badmouth the ex, especially in front of the kids. You also need to develop a plan with your partner. How will you deal with behavioral issues? What are the rules and expectations? Iron these details out early so your partner has your back.
Don't expect to be an instant family. Your stepkids might not love you, or even like you, for a while. In fact, they're probably downright mad that you've entered the picture. It's OK to acknowledge this fact. You can say something like, "I know having me here is hard. You don't have to like me, but you do have to treat me with respect and I'll do the same for you."
Enforce expectations and set boundaries. When your stepchild tries the "my dad lets me do that" line, don't feel guilty or pressured. Simply say, "That's between you and your dad. At our house, this is how we do things." If kids continue to argue, respond with something like, "I'm not going to talk about this anymore," and then walk away. Kids are smart enough to adapt to two sets of rules. Don't let them use this weapon on you.
Build relationships. Initially, you may have to play the role of enforcer, but once kids understand that you mean what you say, it's time to become friends. Spend time together playing board games, reading a book or walking to the park. Go out for ice cream or help your stepchild with homework. Be there when he has a problem or feels sad. As kids begin to bond with you, they're less likely to try to manipulate you because they know you care about them.
Expect to struggle for at least several months, but if your stepkids seem extremely angry even after a few months, get some help from a professional counselor. You'll all be happier if you can talk things out.