I think flying will be your best bet as Portugal is a long ways from just about everywhere.  Start out with www.whichbudget.com to see which low cost airlines fly which routes.   Train might be a good deal in France and Germany — pretty much depends on where in France and Germany you’re planning to go.  From France to England, flying is probably cheapest although you can sometimes get cheap fares on Eurostar, the chunnel train.

An Open jaw (so you don’t have to go back to Portugal) will save time, but might be expensive.  If you don’t get answers here, might repost at least that part of the question on “cheap flights.”

I suspect that there are few cities in the US that will offer direct service to Portugal (Perhaps New York), and if I’m right, you’ll very likely have to fly into London to connect with another flight to Portugal.
In your case, it might make sense to make England the last stop on your trip, and then use London as your arrival and departure city, with a flight from London to Lisbon when you arrive.
As Old Lady has said, Portugal is pretty far from both France and Germany, so you’ll likely need to take a flight from Lisbon to somewhere in those countries, and another flight back to London.

Basie writes:

Not!  TAP Air Portugal has frequent, direct service New York-Lisbon.   I think they may use Newark (EWR) as their POD/POE.  

I agree with OldLady on the ‘Open Jaw’ idea.  When you book your passage you indicate ‘Multiple Destinations;’  Then you fly transAtlantic to Lisbon and return from Frankfurt (or London, or Paris).   Of course you have to book your ticket for the itinerary you have selected.   The ‘Open Jaw’ roundtrip should be substantially less expensive than two one-way tix.

SpanAir code shares with Lufthansa; and SpanAir serves Lisbon.      

But, if the OP doesn’t live near NYC/Newark, then she may have to choose London or Dublin or Paris (as examples) as her arrival city. I haven’t really looked this up, but I suspect that TAP doesn’t serve other American cities.

Only Newark (TAP and Continental), Boston (SATA), and Philly (USAir) offer nonstop service to Portugal.

Newark: TAP and Continental to Lisbon
Boston: SATA to Ponta Delgada (Azores) and Lisbon
Philadelphia: USAir to Lisbon

Last summer, Continental flew to Faro and Porto from Newark, but doesn’t look like they’re repeating for this summer.

Connections with USAir or Continental via their hubs should be very possible if your home airport is served by one or the other. If you happen to live near BOS, then www.azores-express.c… looks like the cheapest option; this airline show Oakland and Providence on their airfare search, too, but only to PDL on Fridays, and not starting ‘til end of June.