Best London airport to travel to continent

Luton is slightly less convenient than other airports because the train doesn't go all the way there and you need to get a shuttle bus from the station. But it's still no big deal.

depends where you are starting from in London and what time the flights are

all the airlines are OK

Heathrow, Gatwick, London City, Luton, Stansted, Southend. If you are coming from central London, they are all similar.
City slightly quicker access, but only takes short haul business flights, unlikely to get a cheap flight.
Heathrow is huge and a mess, takes longer to get through so avoid if an alternative.
All have rail access, but Luton needs a shuttle bus for the "last mile", but that does not add significantly to the time.

Just choose on price and airline, note check-in times, airline rules and luggage allowances before committing.

It is not just airport, airline and cost that need considering, but also flight time and your accommodation location.

I like the smaller airports .
Avoid Heathrow if you can.
Easy Jet TAP Portugal are fine. Easy Jet has its own shuttle bus

https://airporttransfers.easyjet.com/en/transfers

I am surprised that Monarch fly to Lisbon, as they are a holiday airline, rather than serving capital cities. Watch the baggage allowances, which are strictly enforced (size and weight). Easyjet allow a slightly larger piece of hand luggage and have no weight restriction, whereas Monarch enforce a 10 kg limit. Check in luggage is an additional cost.

Do consider the cost of getting to each airport in addition to the airfare.

Prices would have been cheaper if you had booked them months ago, as fares only head in one direction.

I usually do something similar for many of my European trips. Spend a day or two in London and then fly on to my final destination. As previously mentioned try to avoid LHR. I've flown from Gatwick, Stansted, and London city airport. All theee are easily accessible by subway or train. I've used the bus a couple of times and while it was relatively inexpensive it took longer due to traffic. For me the time vs cost trade off wasn't worth it.

I usually fly Easy jet but my friends flew Ryanair last year and thought it was fine. Do read all the fine print and understand that they have fees for practically everything.

I've had good luck with early morning departures. Never had a cancelled flight and I can't remember any significant delays. I figure that with the early morning flights the aircraft arrived the previous evening and if there were any mechanical issues the airline has had all night to potentially resolve it. So a lower likelihood of being delayed or cancelled.