How do I upgrade a ship design without resetting the entire weapons/shields configuration?
Go to the main Menu then to Options then select the Interface tab. The last item on the screen is an option to "Remove Functional Components When Upgrading a Ship Design". Make sure the box for this option is NOT checked and you should be all set.
If I select "upgrade" to ships in the field are they upgraded where they are located?
Yes. However the length of time it will take to perform the upgrade will be dependent on how far away the ship is from your territory. As long as the ship is within your influence area then any upgrade will take 1 week. Be careful because I've seen upgrades take over 100 weeks if far enough away from your area. In that case it's better to move the ship closer to your nearest area of influence which could include a starbase of any kind.
While upgrading, a ships hit points are reduced to 1 and they cannot move until the upgrade is complete so the ship can be very vulnerable. You can only upgrade ships to ships of the same hull size but a cargo hull can be upgraded to a large military hull based ship and vice versa. Finally of course the cheapest way to get a ship of any particular type is to produce it at a planet but of course you could instead just buy it either directly or on a lease of different lengths. Buying a ship is inefficient and wasteful of cash however there are times that it's beneficial to do so. In this case a compromise solution is to build a ship consisting of just the desired hull which is usually quick to produce and then upgrading the hull to the ship's final design. This can save a lot of cash but I forget the actual ratios.
Or should it be saved under a new design?
You can actually save it to the old name but it's safer to choose a new name and the system warns you to do this. If you ignore the warning what you end up in the ships directory is an _0 for the first incarnation and a _1 for the second incarnation without changing the name actually used in-game. It can also get a bit funky if you upgrade the design but don't automatically upgrade every ship of this type that you've ever produced. The best advice is to avoid trouble and just add or increment a number to the ship design name. For my first colony ship I simply call it Colony, and then the next Colony1, Colony2, etc.