In which country is it allowed to practice non-clandestine amateur chemistry?
Your question is basically too broad to be answered in full detail. I suggest that if you want a more detailed answer about a certain part, you should open a question for that matter separately. That being said, I'll try to summarize my own experience as a hobby chemist.

First of all, are you actually studying chemistry? IMHO, if this is within your possibilities, it is the best way to gain experience and also do your your own research. You will have supervision and the best equipment available. You will get the space to do research in your own areas mostly during your graduate studies, but here (The Netherlands) you will also get some space as an undergraduate.

Consider the following points (in no particular order):


You say that finance is not a problem, but do not underestimate the price of all the chemicals and equipment you will need to do any bit of serious chemistry. Scales, vacuum pumps, glassware, solvents, a dedicated fridge, a specially constructed cabinet to store flammables, a fume hood with explosion proof suction, etc (and I'm not even mentioning NMR, mass spectrometers, IR, all of which are available to you as student). To run any decent lab from your backyard be prepared to invest several thousands of dollars (or euros). This can be done of course a lot cheaper, but then you will be limited in your possibilities or you will have to exchange safety for money.
Almost every chemical you will ever need when doing a normal synthesis (i.e. no explosive making or pharmaceuticals) is legally available in most countries (I heard however that some states in the US are quite notorious w.r.t. hobby chemistry, like Texas). You are even able to obtain chemicals that are clearly dangerous and should not be sold to amateurs, like osmium tetroxide, sodium cyanide (not everywhere), gaseous hydrogen chloride in large 110L pressurized tanks. There are a few exceptions:


Common precursors and preprecursors to illicit drugs in large quantities. Most notably phenylacetone and derivatives, piperonal, piperidine, methylamine, benzaldehyde, nitroethane, etc.
Chemicals that can be used to manufacture explosives in large quantities. Examples include: nitromethane and ammonium nitrate.
Chemicals that are severely dangerous to the environment or can be easily abused to poison people, although oddly enough some of the most hazardous are freely available. Examples are: elemental mercury, a lot of (organo) arsenic and other heavy metal compounds, a lot of radioactive compounds, some cyanides (may be available though from mining suppliers), etc.


Note: you are able to obtain most of the above mentioned chemicals (also in larger quantities) if you start a business and have it legitimately registered to the Chamber of Commerce. This business does not actually have to do anything, it can function as a facade. It still won't work for pharmaceutical precursors or very large quantities of chemicals however, but for a hobby chemist that is mostly no problem.
Note 2: At least here in The Netherlands some pharmaceutical precursors and preprecursors can be legally owned and synthesized, they are just not available commercially because of the high change for abuse. These chemicals however (e.g. phenylacetone) can be used in many other legal synthesis and hence it might be useful to synthesize them yourself.
Be prepared to spend a lot of time on this hobby, both studying theory and doing practical work. Independently doing (serious) chemistry is not easy and you will have to rethink and adjust your setup a lot of times. Also the possibility of failure or accidents is higher without supervision. It is not entirely without risk.