Mixing bright warm oranges
When trying to make a bright orange, it really needs to be warm colours without too much blue in them. So you could choose a warm yellow and a mid red or warm red (tending toward orange rather than purple) This will give the truest orange.
If you are mixing cool 'bluish' yellow and red, these colours will contain too much of that third colour (blue), which is the complimentary of orange, and will therefore cancel out the rich orange tones.
Greying/toning colours down
That being said, it is also necessary to know how to mix all of the greyed-down shades, so we can use a beautiful variety of colour and values in our paintings. Small amounts of complimentary colour added to a pure colour will grey a colour down or 'knock it back' and make it less intense.
Making Green Less Vivid
If you look at your green paint and it is too vivid for what you need it for, too 'green', try adding a tiny amount of one of your reds. If it is a bright yellowy (warm) lime green try adding a (cool) crimson. If on the other hand your green is (cool)blueish, you can add a small amount of (warm) scarlet. Only add really tiny amounts of your adjustment colour, and mix it well, before you add any more or you really could make mud. It is so surprising how little you need to add in to alter a mix. TINY amounts really.
Warning! colour mixing can be addictive and fun.
By the way, I recommend mixing paints with a palette knife, not a brush. Mixing paints with a brush pushes paint up into the ferrules where it soon ruins the shape. It is also much easier and quicker to wipe a palette knife clean. You can scrape up a good pile of mixed paint and move it about with your palette knife. It is a good idea to mix up a bit more than you think you'll need, particularly if you are going to mix the final colour with white to make tints.