Choosing a starting colour is not easy, and many designers will tell you this is entirely gut feeling. I partly disagree.
For your first-choice I suggest the strongest, brightest one. The one you will use the least.
You should not really start with picking a colour, you should start asking these questions:
The boring part is that you have to decide what it is for, try to define the use and the users.
- What kind of information are you trying to convey?
- Is it for web?
- is it for use in visualisation? Graphs?
- for engineering drawings and illustrations?
Who will use this?
And not the least:
What background? If print, clearly your background will be white(ish). If for web, the background can be anything.
So let us say it is white, either way.
Then you must decide on how many colours you will need. In general, I would stick with as few as possible, but not one bit fewer :) @S.M points out a few good tools, but as a general example:
You can think of colour as a guide: imagine the way maps use colours. Colours are "way-finding": they should be as consistent as possible, and help "sort content". For example, if you have two identical headings (H1), you want them to be in the same colour and same size.
This is kinda obvious, but you will be surprised at what people do.
As an example taken from cartography, the site Colorbrewer will exemplify a way of thinking that might be helpful. Not saying you should use these colours, but it will give you an idea of one way of using colours.
So, what starting colours?
Say you need four colours. Not everything you colour will be of equal significance, not all will need to be "in-your-face". Sort your content. If this is to be somewhat technical, I would suggest one strong, bright colour, and the rest a little more subtle. A clean orange would be reserved for few but important things (think traffic cones). Do not go with exclusively primary colours: this will make your stuff look like children's toys. Since you are uncertain about this: go with complimentary colours.
You could of course start with any colour, so for your first-choice I suggest the strongest one. The one you will use the least. Colours have connotations, and that depends on culture. As in the west, yellow might signify hospital contagious waste bin bags, the sun; while in the east it might bring connotations of royalty.
Here are some simple examples of complimentary/contrast colours based on a bright orange. As a rule of thumb: if in doubt, go with more subtle colours:
Edit: as of those examples, I might use them but would remove/delete:
number 3 in the first image
number 3 in the second image
number 4 in the third image