The thing with the carriers — well, the West has always been fond of the idea that "The Russians don't build carriers much because they see them as a top cover for the boomers and thus secondary priority", which is quite logical, but not necessarily true. The real reason the USSR has never built a proper CATOBAR carrier is not because they didn't want it or didn't need it, but rather because they couldn't — or, better worded, because they couldn't afford it. The problem is, within the Soviet military the Navy has always been a proverbial redheaded stepson, receiving only a scraps and seconds, especially after the Stalin's death — Uncle Joe has always had a soft spot for the Navy that neither the Baldie, nor the Eyebrowed One had. In fact, Khruschev was suspicious of the military as a whole, and the Navy in particular, but I digress.
The end result is that the Ground Forces, the Air Force and especially the Strategic Missile forces hogged up most of the considerable resource pool that the Soviet military was endowed with, so the Navy brass had to be really considerate with their shipbuilding programs. And we also have to consider that for most of its history the same brass has had only a tertiary say in deciding what they would get. The first say was always on the Old Square (confusingly, despite the name, a street in central Moscow where the CPSU headquarters was located), and the second was the industry, so the brass has always had to negotiate with both of them, which wasn't an easy task to say the least.
So, in short, while the Navy has wanted its own carriers since the 30s at least, there's always been some trouble. They actually almost get approved the construction of the first carrier in 1939, but then there was some little distraction known as WWII, and as the country finally got rebuilt enough to consider upgrading its Navy (not to mention it finally got the Bomb), the major supporter of the Navy, Uncle Joe, has kicked the bucket, and Khruschev was actually cutting the military of all its branches except the missile forces. In Brezhnev times the airs were better, but then the personal matters interfered. The Navy chief of the time, the esteemed Sergei Gorshkov, was actually all for the force projection, but he was a cruiser admiral, and as such poured the Navy's funds into cruiser and destroyer programs, while the rest went to the subs.
This is actually where the "carriers are the cover for subs" myth stems from. It was originally just a justification for continually delaying and cancelling the carrier programs that continued to surface throughout the Sixties and the Seventies. Paradoxically, in the end it even became the tool to push the building at least some carriers through the Party bigwigs, which is where the Kuznetsov and Liaoning (ex-Varyag) come from. You see, the main party ideologue of the time, Mikhail Suslov (a crusty old fella, so dead-set in the olden times of Papa Joe, that his nickname in the Politburo was "The Man in Galoshes" — which went out of style for the 30 years already), held a firm belief that the carriers were "a tool of aggression" and thus unbecoming for the "peace-loving Soviet Union".
So, as you see, the carrier programs were always an uphill battle for the Navy, which had to struggle with its own chief, the indifferent Adm. Gorshkov, the skeptical Defence Minister Mar. Ustinov (a man of industry, he has never held any command except of a militarized artillery plant during the WWII), the openly hostile Suslov, etc. So, despite having completed the preliminary design of a nuclear powered supercarrier in the Sixties (google project 1160), they had to start small, and, as the proverb goes, cut the cat's tail in parts — maybe it'll get used to it. So there were Moskva-class helicopter carriers, Kiev-class VTOL carriers, and, finally, Kuznetsov-class STOBAR carriers which, while not entirely successful, at least were something to write home about.
Only by the very last legs of the USSR the Navy finally pushed through the first true supercarrier design, the project 11347 Ulyanovsk, but the Soviet Union collapsed when the ship was only about 25% complete, and the unfinished hulk remained in Ukraine, together with the 80% finished Varyag, which was later sold to the Chinese. Ulyanovsk was later cut up and sold for scrap, while still-in-trials Kuznetsov had to dash for the Severomorsk lest it would be seized by the new Ukrainian authorities, while the Chinese later completed Varyag and rechristened her Liaoning, after the province where Dalian and Port-Arthur, their main naval bases, are.
The rest we all know — the constant underfunding for twenty-plus years has undermined the Navy and the shipbuilding capabilities to the point that they are struggling even with frigates/destroyers, not to mention carriers. But at least they have started to find their feet under them, so in ten more years — who knows. The shipbuilding assets are getting upgraded, so maybe, maybe. At least there won't be ideological obstacles like Mikhal Andreich anymore.
And, btw, the thing with the nuclear propulsion stems exactly from the capabilities of the industry. While the modern Russia has a healthy steam turbine and nuclear reactors industry, with full know-how and state-of-the-art designs, the gas turbine production was more spread-out in the Soviet times, and much of it (especially a marine propulsion branch) ended in Ukraine. Which wasn't much of a problem until recently, but nowadays this means that it has to be recreated domestically from scratch, which is doable, but expensive ant takes time. Nuclear powerplants, on the contrary, are actually in active production, with a working pipeline, and could be ordered "just now".