I'm going to answer mainly from the perspective of the 1980's since that was really the 'golden era' of the BBS, IMO. The 1990's was more the era of the commercial services (CompuServe, AOL etc. Many of these existed in the 80's but were too expensive/niche for most users) which transitioned to Internet gateways by the mid-90's pretty much killing off the BBS/proprietary use cases.

How was the experience?

Slow. Take the baud rate and divide by 10 to get the characters per second. In the 80's you typically had a 1200 or 2400 baud modem so 120-240 characters per second. PCs were mostly text-based (as was the BBS you dialed into) and had anywhere from 40x20-ish (on 8-bit PCs that could connect to a TV) to 80x25 displays (connected to a 'real' monitor, might require an upgrade on some 8-bit systems) so on a monitor you were looking at no more than 3 lines of text per second. You could just about keep up with reading the text as it downloaded. In the late 80's/early 90's 9600 baud became more widespread a bunch of other improved speeds until finally ~56k (IIRC, this was the theoretical limit of the analog phone system and couldn't often be reached. Also, the higher baud rates assumed some compression which really only performed on some types of content... i.e. not images/archives)

What was the cost involved?

Modems typically cost $100-300 (by the mid-80's) and by the end of the modem era were included with your PC/laptop. In the U.S. in addition to paying your monthly phone bill you typically had zone charges (~$0.10/minute for phone #'s more than a few miles but less than ~30 miles from you you paid to the 'baby bell' who had the regional monopoly in your area) or long distance charges (~$0.30/minute for everything else you typically paid to AT&T until MCI/Sprint became a thing in terms of alternative long distance providers). Most smaller BBS systems were free though the larger ones like The WELL charged fees in addition to any phone company charges. All of the commercial services charged hourly connection fees in those days ~$5-20/hr in addition to phone company charges and if you were lucky, they had a point of presence phone # that was near enough that it would only cost you zone charges. (if you lived in say NYC there were most likely POPs a local call away)

Barrier to entry?

There were a lot of free BBS systems that just required you to set up an account by dialing in and selecting a user id and password. Some the more hacker-oriented systems would require a referral from an existing member. And some people just ran systems for their friends but these were typically private and you'd never hear about them otherwise.

One thing to note re: barrier to entry was often the difficulty of just getting a modem connection to a given BBS. On a smaller BBS, their main incoming line could also be their main voice # so you'd have to call (late) at night or a person might answer. On a more popular BBS, they might have multiple (no more than a handful, typically... each line typically cost ~$20/mo in phone company fees plus the cost of hardware) incoming dedicated modem lines but they were often far fewer than the number of people attempting to connect at peak times so automatic redial in client software quickly became a must. In the mid- to late-90's many people experienced this quite frequently when using AOL for their internet access. Lots of articles and posts at the time about the AOL issue(s).

How similar was it to current social media networks?

I remember the conversations as being more real. Typically the content was more technical since it was mostly computer geeks hanging out on them and we were mostly trying to figure things out / get things done. As the years ticked by, a lot of geek culture stuff started being discussed (i.e. sci fi/fantasy, board gaming etc) It's similar to how if you tune in to ham radio frequencies you'll often hear hams mostly talking about ham stuff. Also, there was a cultural distaste for being overtly/overly commercial if not outright banning it. If someone asked a question and you had a product/service that would help them, it was usually OK to mention it with a disclaimer. If you were mainly there to sell something, you could expect to be booted off sooner rather than later.

How big the industry around it actually become?

Dunno... millions, maybe tens of millions of dollars for the BBS side. For most it was a hobby rather than a business. The commercial services (CompuServe/AOL/etc) side was probably on the order of 100 million+ until it exploded in the mid-90's due to them acting as some of the first ISPs for millions of users.

From a tech point of view what do you think were the major breakthroughs and what made it to the internet we see today?

Very little. Probably the main thing that transitioned over was the concept of user forums and the continued use of .GIF and .ZIP files. The BBS era was a Tower of Babel in terms of protocols (we went from XMODEM to YMODEM to ZMODEM for transfer files... sometimes Kermit which covered terminal emulation as well as file transfer), file formats (.ARC, .ZOO, .ZIP as well as a bunch of others best forgotten) and separation of responsibilities (with the higher speed modems data was compressed but you were also transferring files/archives that were compressed which would result in a slower transfer as the modem tried to compress an already compressed data stream making it larger.) One other annoyance was the fact that there was some platform specific stuff that needed to be address: on the Amiga archives had to be zero-padded either on the sending or receiving end and on the Mac there were resource forks to contend with. So this resulted in a few other oddball things such as .SIT/.SEA files on the Mac. The Internet predated the BBS and was growing exponentially at the same time using TCP/IP since the 70's. FidoNet was basically the BBS attempt to do something similar to the Internet in terms of letting BBS's talk to each other. Some BBS operators fortunate enough to have Internet connectivity were also running gateways and you could dial in to them for UUCP/NNTP to the 'real' Internet.

The reason BBS became a thing was very similar to the reason the PC did: the Internet already existed, had better performance, better reliability, better protocols and better software. However, you had to be connected (i.e. know someone / be associated with certain institutions) and have much more expensive hardware/software well beyond the hobbyist/home budget to use it at the time.