WELCOME ABOARD!
This really is a matter of personal choice at this stage of the game. I'll give my rundown of the current differences, but you should know that all three can be graphics intensive. I'm actually having more trouble with SH3 on my five-year-old setup than I am with SH4. That said, it is my understanding that SH5 does require a dual-core setup that the earlierversions don't. If that's correct and, like me, you don't have that yet, then you will need an upgrade.
SH3 is the oldest title of the series that is still available. On the plus side it is complete, or as complete as it's ever going to get. More modern gamers point out the old-looking graphics. Those of us who played it from the start are used to them. There are many more mod possibilities available for it than the two later releases combined. You can play from the month before the war started right through to the end of the war. There are also a great many things that are not, and may never be fixed with mods. That said, it is a great and fulfilling experience, and has more people talking about their campaign experiences that the later two as well.
SH4 is a great improvement in graphics and crew management, but lacks the mods development that has benefitted SH3. It has many things to recommend it, not the least of which is the fact that it actually runs
better on a medium rig computer than SH3 does. Don't be fooled by the claims that SH4 is for the Pacific Ocean only. The U-Boat Missions add-on (included when you buy the Gold version) is a good start, and the Operation Monsun campaign lets you play the entire war in the Atlantic, so there's a very real choice there. Where SH4 suffers is the lack of small mods, which have so greatly expanded SH3.
SH5 has even better graphics, and as you can see from the screenshots looks amazingly lifelike. New additions include the realistic harbors and the ability to walk through the entire boat and interact more closely with the crew. The bad news is that the interaction is more in the RPG vein and is extremely limited. The worst part is that they (necessarily) had to limit it to the Type VII u-boats, and had to limit the campaign to the early (1939-1943) part of the war. This has drawn complaints from the completists among us, but the players who love the good parts of the game are willing to ignore it, just as SH3 players are willing to overlook the dated graphics of that game for the completeness.
Price of the game itself is no longer a factor, since you can now buy SH5 from Amazon for $16. Whatever your choice is, if you do buy from Amazon be sure to use the link at the upper right corner of this page to get there, so Subsim gets a cut.
I've tried to be fair in my assessment of all three, because as I said there is a very real choice that can be made. I play SH3 and SH4 because of the experience they give me, and as soon as the DRM is removed I will buy SH5 and play it for all it's worth.