It's entirely possible to build up a formidable Soviet Navy. It's handy to have around if you intend to fight Japan earlier than post 43-45 or if you intend to attack the USA/UK at some point. World-class air Craft carriers are out of the question unless you intend to *really* sink a ton of resources into them - and that's kinda pointless lol.
As for the war itself, you get a few interesting options.
1. Submarines. The AI gets *really* irritated if you sink a lot of convoys. It just so happens that Italy and Germany will have a *lot* of convoys ripe for the sinking in areas that reasonably upgraded subs can reach: Norwegian ports, The Baltic Sea, and the whole of the Mediterranean. You also start out with quite a few subs, and torpedoes and AA both upgrade on subs, if nothing else (if you've got the LS to spare) you should upgrade torpedoes and AA guns on your subs, so they'll be relatively more useful. I would also suggest using the tier 3 subs stationed in Vladivostok for this purpose (they're particularly useful based out of Murmansk for raiding German supply convoys to Norway - as they can reach all the way to Oslo). So if nothing else, submarines are a cheap and useful add-on the other nice thing is they give you vision of Axis coastal garrisons, so if you *were* to want to try to do a naval invasion somewhere, the subs can give you a bit of intel. The research here is fairly cheap as well so it conserves LS for land/air stuff.
2. Cruisers. Given that you have a lot of IC and that cruisers are relatively cheap (and you start the game with a naval minister which slows cruiser decay) it's possible to build a formidable force of CLs and/or CAs. By the time you enter the war (if historical) the German navy may or may not consist of only a few ships (+ subs) and the Italian navy will certainly have taken losses, but may still be fairly formidable. If you build *lots* of cruisers you'll easily be able to make enough to completely dominate whatever sea you wish, and also give you cover for naval invasions - and by the time you're ready to fight Japan you should have stupendous amounts of cruisers that will be able to get you safely ashore on the home islands.
3. Battleships. You have enough IC, and start with a *very* small amount of practical in battleships, so it's a viable path to go down. If you start the game by producing another Gangut class BB from day 1, the burst of practical you get will be sufficient to give you 8-9 reasonably priced modern BBs (1938 or a mix of 1936 and 1938 tech levels) by the war's start in 1941, which will give you enough naval firepower to *heavily* influence shore battles, and absolutely smash any naval resistance in any naval theater you wish. If you pair them with CLs, you can sail straight out of the Dardanelles and into the Med and begin ravaging whatever may or may not be left of the Italian fleet (typically if Italy is doing well in Africa, they still have a lot of fleet left in them). Romania also has a token naval force of DDs that you can send to the bottom for fun as well.
4. SH:BB. The absolute cheapest way to get into the surface naval game as the USSR is to go for SH:BBs, because the 1 thing you lack over the other great powers is technology. So obviously, you may want to save your LS for stuff that will be more immediately useful than a surface fleet, and SH:BB are the cheapest way to go, other than subs, in the LS department - especially if you built a *lot* of IC in the opening years, the IC cost of the SH:BBs won't be so bad. The upside for SH:BBs is that you will be able to absolutely destroy *most* enemy naval fleets before they even get close enough to fire until you get into fighting Japan, the UK, or USA. The "problem" with SH:BBs is they are *very* vulnerable to attack. So you *must* make sure they're escorted at all times if there are enemy CAGs or NAVs in the area, or your ships will quickly be reduced to scrap, and that you have an adequate number of screens. I recommend trying SH:BBs at least once (I put 2 in every western naval theater - 2 in Arctic, 2 in the Baltic, and 2 in the Black sea) just for the comedic affect of watching the Dullo or Tirpitz lose 75+% of their strength in a single shot lol.
For example here is a little thought experiment on the costs of SH:BB vs BBs.
In order to get SH:BB you have to spend 2 tech cycles researching the 4 BB relevant techs, (the 1916, and 1936 tech levels) + the SH:BB tech itself.
The SH:BB's sea attack is that of a 1942 tech level BB (which would take you 5 cycles of tech to get, and you would never have prior to 1941), it's convoy attack will *never* be matched by a BB, and *all* of its other statistics, save shore bombardment are *already* outmatched by an equivalent tech BB. So they're a great solution for capital ships to fight Germany and Italy, but will suffer extensively against more modern navies - particularly those with carriers. But you gained absolute naval superiority over the Axis with a savings of 4 tech slots *at the very least*! Stupendous! lol
If you *purely* wanted a navy to do fancy stuff from time to time with (like say.... blockade and invade Norway, invade Italy, and various other places). The cheapest way is just to go with DDs or CLs and hope they can sneak through. But it's *entirely* viable for the USSR to build an *extensive* surface battle-fleet without sacrificing much ni the way of an air force or army as a pure masturbatory exercise - and it can be fun too lol.