This information seemed accurate when published (initially in 2015, then revised several times). However these things sometimes change very quickly, and it's possible some of this is not currently correct.
One place to see movies -especially new ones- is plain old movie theaters ...those places with the huge screen up front and aisles of cushy folding seats. They're not nearly as crowded as they once were (except during school vacation weeks), and many of them have expanded their business model (specialty popcorn, one-time shows, video games, birthday parties, and so forth), but they still exist. And there's a very large, steady stream of new movies from four big studio distributors and a host of smaller distributors competing for their screens.
An ongoing problem for me has been what I perceive as the relative paucity of esoteric/art‑house movie theaters in the North Shore area of Metro-Boston where I live. (The very wide variety of movies available in downtown Boston is inaccessible to me, because at my age and with my problematic health downtown Boston is too far away for me to travel to.) Over the years I've slowly found several esoteric and art‑house theaters and screens, and a theater with very atypical programming. (My own experience [the situation may be very different in other locations] is the art‑house/ grind‑house/ mainstream distinction is more blurry than it was in the past, and it's often no longer useful to entirely include/exclude a particular theater based on this categorization alone.)
(It's true that this area has fewer esoteric movie theaters than downtown Boston and so looks impoverished by comparison. But in reality it's one of the most movie-rich suburban areas in the whole U.S. It's interesting that what I find barely adequate most people consider a dream situation. Either everything is relative, or I'm an incorrigible complainer, or maybe both.)
Of course almost all theaters are devoted to new first-run movies, and never show the old movies that I missed last year or last decade. So I set up my own DVD player too, then upgraded it to a Blu-ray player a few years later. While VideoDisc rental services focus on those who want to see current movies without going to a theater, some of the rental service catalogs are very large and also include lots of the more unusual and older movies that interest me. When setting up my VideoDisc player, I wanted the movies to show on my PC's existing flatscreen monitor, and I didn't have (and don't want) a TV. I had to both make my VideoDisc Player and my PC share a monitor and learn the intricacies of operating my VideoDisc player.
My primary source of borrowed VideoDiscs is the subscription service of Netflix. In the past, the few times they didn't have the VideoDisc I knew existed and wanted to watch, I could usually arrange a one-time rental (not a subscription) from their competitor Blockbuster ...but Blockbuster no longer exists. Most other VideoDisc rental services (Redbox, 3D-BlurayRental.com, etc.) emphasize newer releases so heavily they don't have a large catalog of oldies. I've also occasionally borrowed a VideoDisc from my local library, whose collection is fairly large and eclectic. Other sources for borrowing VideoDiscs in some locations (not here, so I've no personal experience with these) are local chain stores and even occasionally an ISP.
Every so often there's a VideoDisc which I know exists but which I cannot find available for borrowing anywhere. So my only alternative is to purchase those particular VideoDiscs. I've had excellent experiences with purchasing used VideoDiscs, which certainly keeps the cost down. Besides several smaller (often specialized) services, there's the huge catalog available from amazon.com. (Once in a while the material I desire is not the main VideoDisc item but rather is one of the bonus/extra materials. Most places that sell VideoDiscs are extremely bad at finding these; typically only after I learn the placement of the material and the actual title of the VideoDisc from some other source am I able to purchase the right thing.) In any case, thank goodness for real-time computer rough translation and for automatic credit card currency exchange.
I've found that the selection on amazon.co.uk is often wider; among other things films from Eastern Europe that are not available in the U.S. are often available there. Of course (with DVDs moreso than with Blu-rays) this only works for me because I have somewhat uncommon equipment that can play non‑Region‑1 and/or PAL DVDs. This is less of a problem with the newest material, since the PAL/NTSC distinction has gone away for anything with an HDMI cable that uses 1080p resolution, the world is divided into many fewer regions (zones) for Blu-ray than for DVD, and currently about 2/3 of all Blu-ray VideoDiscs are marked region free anyway so they play equally well anywhere.
Streaming services (also often called video‑on‑demand) are quite significant these days. They send the movie over a network as you watch it, so there's no storage and no physical medium (such as a VideoDisc). (The technical aspects of a streaming service are familiar from things like Youtube.) I've almost no firsthand experience with them though. My understanding is there are currently many many streaming services, with a couple of the larger ones being Netflix and Amazon.
Even with the various resources I've located, there are quite a few newly released foreign language films that I don't have an opportunity to see. So I supplemented the art‑house and second‑run theaters and the DVD rentals and purchases with a subscription to one brand new foreign/independent film every month. That subscription to filmmovement sent me a new DVD every month, with both a new feature film and a new short film on it.
Fortunately several local film festivals have sprung into existence relatively recently. The Northern-Lights/Newburyport-Documentary Film Festival has existed for over a decade now. I initially became aware of them because one of their venues is one of the small movie theaters I frequent. The Manhattan Short Film Festival has recently added a local venue. And the Salem Documentary Film Festival has grown in only the handful of years in which it's been in existence to be one of the largest all‑documentary film festivals in the whole country.