Campbell began by imitating Lovecraft, and he still does the occasional Lovecraftian story (including his fairly recent The Darkest Part of the Woods); but he has long since gone his own way, with an ability to evoke urban horror that Lovecraft would not have been able to achieve, in my opinion. This is not to say his work isn't supernatural -- it often is, or hints at such; but that his settings are very contemporary and rooted firmly in the urban milieu he knows best.
As for whose work his most resembles at this point... I suppose the closest would be Robert Aickman; but there are strong differences, as well.
Where to start? That's difficult to say. Campbell is quite prolific, and not everything he has written is of the best quality; but a substantial amount is very good indeed. Nor have I read everything he has written (though I hope to remedy that over time). So my suggestions will have to come with the caveats that I may be missing some things worthwhile....
Collections:
If you can find it, get hold of a copy of his transitional collection, Demons by Daylight (1973). It has a piece or two of Lovecraftian horror, but already very much showing Campbell's own voice; while there are quite a few pieces of a more experimental and modernistic nature which are almost unclassifiable. I would also suggest Dark Companions (1982), Scared Stiff: Tales of Sex and Death (1987); Alone with the Horrors (a retrospective overview; 1993); or Ghosts and Grisly Things (1998). All of these will give you a good sampling of Campbell's variety in the shorter forms, and most are either in print or have fairly recent printings, and tend to be quite inexpensive (in the main). Or, if you'd like a glimpse of his shorter Lovecraftian writings throughout his career, go for Cold Print (expanded edition, 1993), which collects together many of these.
Novels:
The Face that Must Die (I'd suggest an edition post-1983, when the missing text was restored); The Parasite (1980); The Hungry Moon (1986); Midnight Sun (1990); Needing Ghosts (1990); The House on Nazareth Hill (1996); The Darkest Part of the Woods (2003); The Overnight (2004); or The Grin of the Dark (2007).
Cambell isn't for everyone; his ambiguity is a sticking point with some readers; but I'd say he's one of the most literate, subtle, and powerful writers working in the field of the modern weird tale....