It's true that various parenting styles can mesh or clash with individual children's native "style." And child development experts now believe that television is a no-no for kids under two, and should be extremely limited for older toddlers, because it does mess with their little brains and attention spans, and it limits their ability to imagine their own stories and mental pictures, so it's good you're asking these questions while your guy is still a "little."
Some 2yo's take to structured activity and can sit still, if needed, for maybe 10-15 minutes at a time. Some simply can't. I don't think my grandson could until he was around 4 unless he was really tired, or had chosen the focus of the moment for himself. But he was/is a great kid, generally involved and cooperative in spite of the occasional juvenile meltdown.
Does your son take general direction well? Will he take your hand crossing streets when you tell him to? Come when you call? Pay attention if you tell him no? Does he communicate clearly, with words and gestures? Can he stay focused on activities of his choosing?
You only give this one example, so it's really hard to give much specific feedback. But let me give you my list of favorite tips for working with this age group. In spite of plenty of behavior that runs contrary to what Mom and Dad want, your son is not "trying" to be naughty. Children truly don't want to cause problems or get in trouble if they have any other way at all to meet their own growing need for autonomy, independence, and control of their circumstances.
We busy, over-scheduled grownups want everything to flow according to our adult expectations, and have a lifetime of practice learning how to make that work. Your toddler doesn't know any of that yet, and he will probably have a couple more years in which he just won't see very much from your point of view. He can't help it. Dawdling, digging in heels, and even tantrums are a natural outcome of becoming more frustrated than he's able to endure. Occasional stresses like travel and changing schedules, or illness, or any major change, may contribute.
Though your toddler won't ever behave like an adult and make all the choices you would make, there are tried and true methods for eliminating much of the mutual frustration the two of you will experience during the next couple of years. While some steps require a bit of advance planning or extra patience from you, overall, you'll spend less time and energy than you would if dealing with behavior that isn't evolving in a positive direction.
1. When he wants something, empathize, big time, and in the child's language. I love the advice of Dr. Harvey Karp on how to get on a tantruming toddler's wavelength in this and several related videos: http://www.youtube.com/watch?v=G6KnVPUdEgQ&feature=re... . Once your little guy realizes you do care about what he wants, he's more likely to be able to calm down and cooperate with what you need from him. And there are lots of positive ways to approach this, rather than just saying no. They hear NO! so often, and they can be so frustrated. So look for ways of finding a mutual "yes," and save "no" for those occasions when children are trying something dangerous.
2. Keep it playful, and keep it respectful. Children learn primarily through play and imitation. You may need help with this if your parents didn't model a lighthearted and kind approach when you were little. The book Playful Parenting is a great resource. Your son's sense of humor is developing, crude though it may be.
There will also be times when you must have cooperation, like diapering or trips to the store, so keep a special toy or distraction that he gets to play with only at those times.
3. Here's a big one: give advance notice when you'll want him to be doing anything differently, especially when he's grooving on some activity/play. Children absolutely hate unexpected transitions. With my grandson, we let him know at least a couple of times that a change is coming ("We're going to go to the store / have lunch / take a nap pretty soon." … and then a second alert a minute before making the change.)
4. Whatever the next activity is, make it sound as desirable as possible. Give some detail about the ride in the car, or a favorite food at lunch, or sweet snuggles before nap. Be positive and enthusiastic. And be calm. Desperation will show, and even though he's too young to deliberately engage in a planned power struggle, he will quite naturally sense any cracks in your resolve. And this is where future power struggles truly can begin.
5. Try to keep demands low when he's tired, over-managed, hungry, or sick. He won't have any emotional reserves left with which to cooperate.
6. Get to know his most likely trouble-spots, and plan ahead. For many kids, it's when they want some tempting object or food they've seen. So keep those things out of sight when possible. Be prepared with a distraction – for example, another toy he likes when you have to take ______ away, or a healthy treat when he wants a junky snack. Laughter, introducing a new game, a few twirls and bounces, hugs or tickles, a goofy song, can help break into his determination to get something he wants (that short attention span is both a curse and a blessing).
7. Avoid bribes, but let him work toward occasional rewards. Don't try to buy cooperation with "IF you'll do X, we'll let you have Y." Instead, phrase it as if he gets to assist in advancing something good for himself: "Hey, as soon as you help me get X done, then we get to do Y." It sounds like a small distinction, but it's important. It gives the child a chance to cooperate in what feels like choice, even when there's no "if" about it.
8. Be sure he gets lots and lots of physical activity during the day. Time outdoors in nature is calming for most children. If she has pent-up energy, it will have to come out some way, and unless channeled in a positive direction, it will likely to add to both your stress and hers. And limit TV time – the passive receptivity to all that fast editing seriously interferes with children's normal brainwave patterns, making focus and engagement in the "real" world more difficult.
9. Whatever you are trying to teach will need to be repeated hundreds of times over the next few years. That's completely normal. Attention spans and impulse control are extremely limited in toddlers. (Look at your parenting contract, and you'll find it in microscopically-tiny print under "I agree to the following terms and sacrifices.")
In your case, I would add LIMIT TV and read to him, short books that can help him develop his imagination. When he does watch, maybe up to one hour per day max, try to interact with him, making comments about the show and asking him questions to deepen and enrich his total experience. And talk to him about everything you do, explaining what this is and how it works, why you are doing that, what's coming next, and how you are feeling.
Spanking and time-outs don't work for every kid, and will sometimes actually backfire over time. Especially with spanking: children may be scared, shocked, or shamed into compliance, but behaving for the sake of avoiding pain isn't the same thing as developing an internalized sense of "good." Consistent, calm guidance and modeling what you DO want from him will work better in the long run than punishing for what you DON'T want him to do. See more on this by googling The Science of Parenting or Emotion Coaching.
I wish you both well. Pay attention to what you love and appreciate about your son, and make sure he knows. Treat him with calm and respectful authority. Children crave attention and approval, and if he knows you're noticing his good moments, he'll try to create more of them.