[AntEater]

Currently, the russian navy has managed to make operational the remaining ships of the soviet navy. The Kuznetsov and the remaining Kirovs and stuff are operational again, and regularly putting to sea.
Also, the Black Sea fleet was able to sortie a large number of vessels and fulfill its (relatively easy) missions in the georgian war.
Given that the Black Sea fleet is the oldest on average, this must mean that readiness really improved. Especially since the same fleet was only able to send one intelligence trawler to the Adriatic during the Kosovo war. All other ships were down or broke down in the black sea in 1999.
However, newly build ships are rare yet, and those that are relatively new are mostly one-offs with the remained of the class cancelled.
Serviceability must not be great. Most ships, except for the black sea fleet, are not that old (mid-late 1980s mostly) but the years of neglect in the Yeltsin era were certainly not good for them.
Warships, especially with turbine propulsion, are complicated machines, even with the russian talent of keeping things fool-proof and simple.
On the other hand, the russians were able to sail a mid-1960s Kashin class destroyer from Sevastopol to the georgian coast on a day's notice, so things are looking up for older ships as well.

New ships are slowly becoming available, like the Steregushy class corvettes/frigates.
A new class of guided missile destroyers (called frigates), the Gorskov class, is apparently authorized and under construction. Those will be based on the ships build for the indian navy. With 20 planned, they will not represent a real increase in ship numbers, as they will replace both destroyers and frigates.
But these will be highly modern DDGs with very good missiles (and typically russian, lots of them!).
There's no program to replace the larger ships yet, but apparently the russians are considering to build new carriers in about ten years.

Re submarines, I don't know how their readiness, but things seem to have improved much since the Kursk. New submarines are actually being build instead of just being announced.
Interestingly, the priority is SSBNs, with new SSNs coming second only.
The russians will not go all-nuclear like the french and british, most likely due to the value of the SSK for coastal defense but also because SSKs are a lucrative export market.

So today, I think the picture is:
The russians have an active navy again, which cannot be compared to the soviet navy in size but still dwarfs many regional navies in capability and retains all major types of combat ships.
The replacement program of the russian navy seems to aim at preserving the current size of force while replacing the cold war era ships with modern ones.
Russian shipbuilding managed to survive the cold war relatively intact due to exports to India, Vietnam etc. In fact, thanks to globalization, LBOs, bancrupcies, idiotic defense policies and cutbacks, the russian shipbuilding industry is in no worse shape than its european or US counterparts.