What kinds of things are they saying no to - what kinds of things do you need to get them to do?
Janet Lansbury has some good advice for this age I think. Try her blog/podcast. I love her concept of "confident momentum" and being "unruffled". The other book which is great for this age is How To Talk So Little Kids Will Listen. There is an app which goes along with it which will let you remember the main points in the moment. (Way back when on MN, we used to have an A4 crib sheet people printed out and stuck to the fridge!)
One important thing I did learn when babysitting as a teenager was to avoid using questions when you're giving instructions. Don't say things like "Right, shall we get ready for bed now?" or "Can you get into the bath for me, poppet?" to a 3yo because that sounds like you're presenting it as an option that they can say no to. Adults often give instructions like this to other adults because it sounds polite, and adults are aware of that and understand it would be rude to outright refuse without a good reason. But young children don't have that grounding of the social niceties, so will take it literally. Don't be afraid to be direct, they prefer it when you are because it's clearer.
If it's not really an option but actually an instruction, present it as one and don't use a question. "Time to get ready for bed!" "Time to get into the bath!"
They will feed a lot off your energy, so if you sound enthusiastic and interested about the activity, they often will be as well. Whereas if you sound annoyed and impatient, it often makes them grumpy and more likely to refuse. If you sound unsure and dithery, they don't like that at all and it winds them up and makes them anxious so their behaviour becomes quite difficult to manage. If you don't know the tone of voice I'm talking about, watch any children's TV programme and essentially try to emulate the way that children's TV presenters talk. Young children tend to be very responsive to this way of speaking. You can tone it down once you've got the hang of it but if you're not sure, try really getting into the role initially to see how it feels.
You can also try a closed choice. "Do you want to wear your Thomas pyjamas or the Paw Patrol ones?" "Which story do you want to read today after you've brushed your teeth?" Note the question is not do you want to get into your PJs? But Which PJs do you want? So the instruction (You have to put on PJs) is included in the question.
You can also add play/silliness to an instruction to make it more fun for them. So for example help them get changed by being a getting-dressed-helper-robot, who is occasionally broken and doesn't follow instructions properly. Or when brushing their teeth, look for colours or animals hiding in their mouth, or sing a teeth brushing song. When eating be dinosaurs eating tiny brocolli trees. When it's time to tidy up, make it into a race, a challenge or put on a tidying up song and dance while doing it.
Use first/then when appropriate e.g. "First we put on our sun hats, then we can go out to play." This is usually more effective than the same instruction made into a threat e.g. "If you don't put on your sun hat, you can't play." The double negation is also a LOT for a 3yo to process in terms of language, and they often won't understand. First/then is much clearer.
For most of these tips, they work best when everyone is in a good mood already. You won't have much luck if you only remember to use them after a tantrum has begun, although sometimes you can diffuse a bad mood with silliness. Try to stay ahead of tantrums by ensuring there are frequent opportunities and reminders to to eat, drink, use the toilet (and if the child is prone to saying they don't need to go and then having an urgent need later on, don't ask them if they need to go, tell them to have a try) and a balance of sedentary/calm activities and more active ones - keep an eye on what their energy levels are doing. If they are getting too silly for sit-down things like board games or drawing, get them outside or doing something active. If they are getting too hyped up or overexcited with active games, try redirecting to something calmer like colouring or listening to a story, and intersperse more full on days with some downtime so they don't get overtired and go over the top.
Keep something up your sleeve as an emergency distraction and don't use it any other time, in order to protect its value. It literally doesn't matter what it is. When I was a child I remember having a babysitter who had this special game she used to play with us very occasionally. We would get wildly excited and do almost anything in order to get to play this game. The game was literally just her lying on the sofa with one leg held at an angle in the air. We would pull the leg down and let go and she would say "DOOOOOOIIIING!!!" and bounce it back up. Honestly, children are very easily entertained if you keep something scarce 
another game my children love is the sausage roll game, where we take the duvet off their bed, put it on the floor, they lie across the short end and we roll it up and then say they are a sausage roll and pretend to cut them up and eat them. Hilarity ensues.
If children are winding each other up and feeding off each other, find a way to separate them while you work out a plan. Give one of them a special job/task/mission, give them each a distraction in a separate space, or strike a deal with the older or more sensible one to cooperate in exchange for some kind of reward or high value activity while you focus on the younger or wilder one without the distraction of the other child winding them up.
The good news is that all of these tips are also way more effective on other people's children