I am new to Russian language.
What гласные are not allowed after к, г, х and ж, ш, ч, щ?
And what гласные are allowed in this case?
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ы is never used after ж, ш, щ, ч and only used after ц in several roots (цыплёнок, цыган, цыпка etc.) and on morpheme boundaries: огурцы, куцый etc.
ю and э are only used after ж, ш, щ, ч, ц in several loanwords and transcriptions: жюри, жюльен, брошюра, парашют, пшют, амбушюр, Жюль, Чюрлёнис, Шютц, Шэрон, Шэньчжэнь etc.
Iotized back vowels (я, ё, ю) are mostly used in loanwords after к, г, х: закят, кёрлинг, гюрза, etc. However, Russian tends to cancel palatalization after velars, so vernacular forms like пекёт, жгёт, могёт are used more and more frequently, and the form ткёт (instead of obsolete *тчёт) even made it into the literary norm. Unlike other Slavic languages, е and и are pretty common after velars: на ноге, на руке (compare Belarusian на назе, на руцэ).
э and ы, on the other hand, are almost never used after velars (к, г, х) in Russian, except in some loanwords (гэльский, хычин, гымза etc.) and several onomatopoeic words (хых, гы, кыш and similar). Old Russian had it the other way around (velars were always hard before the front vowels), and some remnants of this can still be observed in old Moscow pronunciation (traditionally used on stage in Moscow theaters), which uses ы after velars on morpheme boundaries: русскый, долгый etc.
In speech, ж, ш, ц are always hard (non-palatalized), while щ and ч are always soft (palatalized) in Russian. Hence, all differences in writing are purely etymological: there are no differences at all between the first syllables in цыпка and цирк; жёлоб and жопа; щётка and Щорс; and so on.
Also, э in Russian is quite uncommon after hard consonants, if the consonant can be soft at all. So it's only used in writing for several loanwords and transcriptions (мэр, сэр, пэр, мэтр, рэп, Сэм etc., mainly those that can be confused with corresponding soft consonant counterparts), and a little bit more widely in speech (тест, секс, кафе, термос, Дели are all pronounced with a hard consonant before е, as if it were an э).
Russian, though, tends to soften pronunciation of hard consonants before е: шинель, рельсы, термин are soft in the literary norm; энергия, компьютер and several others are on their way there.
However, ж, ш, ц are never soft in Russian, so, despite the fact э is almost never written after those letters, they are still pronounced hard.