Best places as base for many day trips ?

As mph said, your question is too vague. Narrow it down. Where have you already been? Which countries interest you? If you've read all the RS books, you must have a good idea of where you want to go.

As for your budget, I think it will be hard to impossible to get rooms for €50 a night. You can get dorm rooms for less, but then where are you going to put your laundry to dry?

Sorry I did not specify areas I wish to visit. As I plan to travel for around 4 months extendable to 5 and my return ticket is not bought yet, I am flexible on the places but I wish and try to visit most countries.
My rough idea at the moment, based on RS books:
Britain: 3 weeks less London.
Scandinavia: 3 weeks
Switzerland: 2 weeks
Italy: about 3 weeks, less Rome and Venice
Greek Islands: I week
Central, Western, Eastern Europe: 8 weeks, less Spain, Portugal, Turkey, Paris and Berlin.
I do find some rooms recommended by Rick Steeves for less than 50 euros but my editions are mostly around 2010.
Thanks.

I don't see you as a troll, it is a very reasonable question.

How are you handling your visa difficulties? If you are eligible for the Schengen visa waiver program you are limited to 90 days in any 180 in the Schengen 26 country area (most of Europe).

You have listed 119 days in the Schengen area (17 weeks), which cause you to be an overstayer for 29 days - that could have serious consequences to both your wallet and your personal freedom.

Thank you for pointing out on the Schengen countries. I will limit it down to 90 days. I don't need any visa.

I think since you're talking about what you call secondary cities, not the 'biggies', then you will probably be able to find budget hotels or b&b's for around €50 or less/night. I traveled around central Europe last summer and rarely spent over €50/night (my most expensive place was €80 in Brussels), all were small hotels or apartments, no hostel beds. My apartments in Budapest & Prague were both <€50/night, and they were nice, safe, centrally located places. This was for a solo traveler in June & July. They're out there you just need to search for them and it helps if you reserve ahead rather than try to find places on the fly. Since you have such a long time you should be able to stay long enough in each place to make an apartment a good use of your accommodation funds.

You still need to do your own research based on your interests as to where you want to go, then come back for recommendations for each area. Any town on a main rail line is likely to have day trip opportunities to other places so you can stay put for longer periods of time in one location and save the time and money of relocating from place to place so often.

ps: don't rely just on RS guide books for your research, branch out to other sources.

You don't need to follow Rick's two-night advice at every stop. It sounds like you can be flexible on this trip, so just decide as you go. You will sometimes end up staying an "extra" night when you do want to do laundry, or sleep in, or the weather doesn't cooperate, or you find out about a neat event/side trip from the tourist office or B&B owner, or you find a really good-value accommodation, or you just didn't get all your sightseeing done yesterday. I would also agree that many smaller towns will have single rooms for not more than 50 euros.

Do update your guidebooks once you narrow down your list of countries, but noting publications dates or next-arrival dates on each book page in the Travel Store. For instance, the Italy 2015 edition is already out, but the new Scandinavia edition won't be in stores until March. Note that the England book covers several towns that are not in the Great Britain book; the Provence/Riviera book covers areas not in the big France book; and the Croatia/Slovenia book covers those areas more thoroughly than the general Eastern Europe book.

Any time that you spend in Britain does not count toward the 90-days Schengen rule.

Thank you for all your replies.
I apologise if I missed out anything as I am moving around and depending on my android phone, I can't see and refer to every reply as I type.
I am restricted by the Schengen rule. Initially I thought I can spend some days in Schengen, move out to Britain and return back to Schengen will be able to stay for more than 90 days count. I can't do that as my total days in the whole trip is less than 180 days so my Schengen days in total cannot be more than 90 days; and in order not to spend more and if I get the Eurail 90 days global pass, it will be more cost effective to leave Oslo as soon as I arrive, travel the non-Schengen countries and return to do Schengen and travel for 89 days. With a limit of only 90 days, looks like I do not have much chance of staying anywhere for more than 3 days due to my too many must-see places. Right now, I can only think of Florence or Sienna and somewhere around or at Lucerne. I will work out on a more certain itinerary and put up another post and hope you can advise again.
Thank you again.

There is a fantastic youth hostel in Taormina, Sicily and so much to do in Sicily.

In the Country of Luxembourg they have very modern youth hostel accommodation.