>>719109250
Burst or Burst ReNewal (PC, avoid the PS4 version) depending on which one is easier for you to access, then Deep Crimson or Shinovi Versus, then Estival Versus (or Deep Crimson), then Peach Beach Splash. Burst and Deep Crimson are generally considered to have the best gameplay and story. Estival Versus and Peach Beach Splash have the most ecchi/fanservice.

>>713957073
From Steam, the best to start with is probably Burst Re;Newal as it's chronologically (a remake of) the first game in the series and sets up the characters and setting.
>Do I need the DLC?
Probably not, but you can grab the character addition DLCs as a start. The rest are mostly fluff.

>>713553130
I barely ever used the 3D mode on my 3DS. The main draw of the 3DS games over the later entries was the different gameplay, the more serious stories and the focus on the rivalry between the two original schools allowing for character building. There's no Intimacy Mode, instead there's a more basic Photo Mode where you can dress up the girls and have them strike poses so you can take pictures.