Can I get cheaper journeys by avoiding flying?

For some European locations National Express isn't a terrible option - the selection of destinations isn't huge at, well, four (and they currently aren't available but should be again at some point) but the cost is probably the closest to your cheap flight target at around £10-20 each way.

A more expensive but far more flexible option would be an Interrail pass - at £157 for a 4 travel days-in-a-month pass it's obviously more than the flights, but it can get you to most places in Europe.

@mdewey's comment about the reservations and Interrail made me check soemthing - the Eurostar has a mandatory reservation fee of €30 London - Paris or €35 London - Amsterdam, since the OP is coming from London that's going to rather significantly increase the cost of the Interrail option! Even if they manage to avoid reservations for the rest of the journey that pushes the cost to over £200!

This brings me back to the coach idea - if booked far enough in advance Flixbus can be reasonably cost effective (probably < £50 each way) at the cost of limited destinations and long journey times.

I can recommend the Rome2Rio website. It will give you lots of options for getting between two places. This includes flights, trains, long distance coaches, local buses, and all sorts of other options.

You say you live in London and mentioned Sweden as a possible destination, so I asked it about routes between London and Stockholm. At the present time, the cheapest option it gives is flying from Stansted to Arlanda with Ryanair. Including transport to and from the airport, it reckons the cost starts at £59. This price assumes you're flexible enough with times to actually get the best fares, and don't need to pay for extras like hold luggage.

The site quotes prices starting at £182 for travelling the whole way by train. This includes a sleeper from Hamburg to Stockholm, so when you compare prices, you should take into account the fact that you don't need accommodation that night. The most expensive leg by far is the Eurostar to Amsterdam, with prices starting at £110 for that leg alone. (Edit: that's according to Rome2Rio; Willeke points out below that you can get significantly cheaper on the Eurostar website at the right times.) This is unquestionably more expensive than flying, though in my experience you're more likely to get the lowest quoted figure on trains than when flying.

If that's is too expensive, another Rome2Rio gives an option to get the whole way by coach, with prices starting at £66. That's a very similar price to flying, and may work out cheaper when you factor in the Ryanair's hidden extras and difficulty getting the headline promotional rates.

Personally, I'd hate the idea of spending 36 hours on coaches, so I might investigate using a coach to get to Belgium or the Netherlands, and then trains onwards. Rome2Rio isn't very good with these mixed-mode options, but you can always search two separate journeys – e.g. London to Brussels, and then Brussels to Stockholm. This tells me I can get a bus from London to Brussels for as little as £10, either with Flixbus or BlaBlaBus, and then trains onwards to Stockholm (including a sleeper), with prices starting at £91. This gives a total of £101. That may be a useful compromise between cost and comfort.

Rome2Rio is great for getting ideas and approximate prices, but you'll want to look at the options in more detail on the train and coach operators' websites. You may find its hard to get tickets at the cheapest advertised prices. I also find the Deutsche Bahn website useful for planning rail journeys in Europe, though it's not great for prices outside Germany.

If you are concerned with cost and carbon footprint you can hitch-hike (local laws permitting).

A Channel crossing by P&O ferry for a foot passenger is about £12 or £13 today one-way. Or you can hitch-hike across the Channel, please see Hitchwiki.

With hitch-hiking you aren't creating any additional journeys.

Also, it depends on the purpose of the journey. For me, getting there is just as much a part of the journey as the destination (travelling) and I don't like staying in one place for more than a couple of days.

Outside Europe, hitch-hiking might not be viable, but there are cheap forms of shared transport such as train and bus services, and on-demand minibuses, shared taxis or jeeps. If you travel the way local people travel, it is affordable, and you'll get a much closer contact with the country than if you see it from a train or hotel window.