Travel from London to Spain?

Looking at skyscanner.com for a random date in May, I see numerous nonstop flights from various London airports to Barcelona, with about 2+ hours flight time. Fares on Ryanair and Easyjet are shown as low as $17, though some of those are from airports like Stansted and Luton that are not as convenient to London as Gatwick and Heathrow. Also note that the "budget" carriers charge extra for almost anything they can -- seat selection, baggage, food, etc. British Air and Iberia also fly between London and Barcelona for slightly higher fares.

The Spain and Portugal forums are the best places for specific questions about what to do on the ground in those countries. And, in planning your overall trip, consider "open-jaw" flights from home to London and from Spain or Portugal back home, so you won't spend time and money backtracking. These itineraries can be found under "multi-city" on search engines and airline websites, and fares are usually comparable to round trips.

there's a website called Rome2rio that will lay out your various travel options. Flying seems obvious, but you might want to use some train and/or some car.

Fly from London to Barcelona - it’s easier from Gatwick, Heathrow or Stansted. Luton is further out. Luggage limits have changed with the airlines, so check what is and isn’t permitted before you book. It’s cheaper to check an oversized bag at the time of booking than at the gate.

As with all advice in the past two years, I would try to limit the number of countries visited to avoid getting in a tangle about the various Covid restrictions.

How long is your trip? Presumably you are having a week in London (which will barely scratch the surface), a transit day, then 4 nights in Barcelona, half a day to transfer to Valencia by train, then 3 nights there. Getting from Valencia to Seville - there aren’t any direct trains, so taking a flight would be the best option - Iberia or Ryanair are your best options. Seville ideally needs 3 full days/4 nights there.

Then the tricky bit - Spain and Portugal are not well connected by public transport. Your only flight option is TAP (also known as Take Another Plane). Cascais and Sintra can be seen from Lisbon as day trips, so a week in Lisbon would be sensible.

All of London’s surrounding airports are easily accessible. Luton may be further out but it’s only 35 mins from St Pancras – Stansted is an hour from Liverpool Street; Gatwick 35 mins from Victoria. Heathrow is anything from 15 mins on the train to nowhere and an hour on the Tube to somewhere.

For rail connections, study www.seat61.com

We do this every time. No matter where we go in Europe, we usually start in London. From there, you can get cheap flights on Ryanair or EasyJet anywhere you want to go in Europe. Ryanair gets a lot of flack from some people, but its always been reliable for me and its really cheap.

These flights usually go from Stansted, Gatwick or Luton. Stansted and Gatwick have easy direct trains from central London direct to the terminal (Stansted Express and Gatwick Express, plus regular local services). Luton is a bit more complicated as there is no station in the airport, so you have to take a shuttle bus from Luton Airport Parkway Station.

How many nights are you able to spend in Europe, not counting the night on the plane? Your destinations are going to mean at least two flights in the middle of the trip, I think, and I find mid-trip flights really annoying and disruptive. You have to allow so much extra time at airports these days, and to me it just breaks the relaxing mood I like to be in.

The train route from Valencia to Sevilla involves a change of train in Madrid, which is likely to mean some layover time. Still, I'd prefer that to dealing with yet another airport.

Valencia, though a pleasant place, is quite an odd choice for a first trip to Spain, much less a first trip to Europe. Do you have a personal reason for choosing it rather than Toledo, Cordoba, Granada, Madrid?

Ground connections between Spain and Portugal are limited and awkward. Your best bet will be to fly, I'm sorry to say. TAP has non-stop flights from Seville to Lisbon, though I don't know that they're available every day of the week. I think RyanAir may fly that route also.

We often discourage folks from trying to combine Spain and Portugal on a short trip because of the logistical challenge, which is one reason I'm asking how much time you actually have. There's a great deal of variety in Spain. I do like Lisbon, but it would be simpler if you substituted another destination in Spain.

Skyscanner.com is a good source for checking intra-European flight options.

Thank you to everyone for the replies! They've all been really helpful. 
To answer your questions: We're going for about 2 1/2 weeks (that's as long as we can get away). We're going with a friend who has lodging for a week through a home exchange in Denia, just south of Valencia, so we're sort of building the trip around that week. We're planning to use Denia as a home base for the week we'll be there and do day trips in the area. 
Also, based on your suggestions, we're going to eliminate the Portugal part of the trip and spend that time exploring more of Spain instead. And it looks like our best option for getting from London to Seville (we've switched the order of cities around a bit) will likely be Ryan Air. Thanks again to everyone who responded. ALL the information was helpful, and I'll be using the recommended sources and websites as we plan the rest of our trip. :-)