When I drive by houses doing exactly what you are planning, sometimes I cringe at what I see happening. There are many different types of board insulation and it’s very easy to add the wrong type, or probably a surprise to some, but the wrong amount.
Some rigid board insulations are a vapour barrier, some are not. Some are air barriers, some are not. XPS foam insulation at about 1 inch thick will act as a vapour barrier, EPS foam insulation will start to act as one at about 2 inches, and polyisocyanurate (the one with the foil on both sides) is a vapour barrier at any thickness (foil is waterproof). You can read all about rigid insulation panels here.
Installing insulation that acts as a vapour barrier on the outside will not serve you well, except if you add enough of it, to the point where about 2/3rds of the total insulation is on the outside. At that point what you are doing is building a R.E.M.O.T.E wall, read more about that here. What that means, is if you have 3.5 inches of fiberglass in the stud cavities, you can add about 7 inches of EPS foam on the outside with no problem, because you will then be keeping the interior of the wall warm enough that the ‘dew point’ is no longer an issue. We would suggest EPS because of the plastic based foams this is the least problematic environmentally. Here's a link to a product that is simple solution to achieve this, it is an EPS foam panel with a metal rail embedded in the center for exterior insulation that is used to attach it to the wall and that the strapping is then attached to for installing siding. You would just screw it to the wall, then screw strapping to the same panel for installing future siding, which sure makes the job quicker.
If you prefer a wood solution over plastic (though EPS is really mostly air), what you could also do is use a Larsen truss wall system as a double stud wall, (see more here) which are ideal for improving wall insulation when renovating homes & for building new homes with high performance insulation systems such as Passive House or LEED. You can then either fill the new cavity with dense packed cellulose insulation, or even consider the latest kid on the block, natural based hemp insulation products.
What is really important is that you don’t affect the wall’s ability to dry. For that reason if you wanted a simple "one size fits all" exterior insulation skin option but don't want or aren't able to add 7" to the exterior of your home, we would recommend Rockwool rigid board insulation (known as Roxul prior to re-branding). It’s sort of a failsafe insulation in the sense that air goes through it, moisture goes through it and water drains through it. You can add as much rock wool as you want, or as little as you have to if there are other considerations like property line setbacks or such, because it won't alter the basic function of the existing wall.
After that, install a vapour permeable drainage plane, and vertical strapping to allow moisture to vent and water to drain, which won’t happen if you install strapping horizontally. We have a page on how to install siding here where that is all explained in detail.