There are 3 options:
1) cross a separate CMS with a separate forum engine;
2) take a CMS with a built-in forum (or where the forum is in the form of a separate module)
3) take a forum with a built-in CMS (or where the CMS is in the form of a separate module)
The first option is for a good programmer. There are ready-made solutions, it may not be so difficult to do, but if you update the engines, refinement will be required. Yes, and a lot of headaches.
Two other options - depending on what you need - a powerful CMS and a simple forum or a powerful forum and a simple CMS.
The most popular engines: Vbulletin, IPB, phpBB, perhaps even SMF, Xenforo can be called. phpBB, SMF are free.
I don't know about IPB, but Vbulletin 4 has a built-in CMS. Comments on the news become posts on the forum. It is quite convenient, although many scold this version of the engine. In principle, for any popular forum (Vbulletin, IPB, phpBB - for sure) you can use the CMS module.
The design for the forum is done in the same way as for the CMS. If you make a good design from scratch, there will be a lot of work: all sorts of buttons, graphics, a bunch of styles. Then all this is still to make up and integrate into the forum.
For money. Put-configure the engine + attach a template design will be the same 3000-5000r. Plus the cost of the engine, if it is paid. Here, making a design from scratch, making up, integrating - will be much more expensive. And any normal forum will also require the installation of various hacks and modifications.
That's what I would do if I were you. I would try different forum engines (see above), look at what CMS are included in the kit / separate modules, look for examples, climb as a user and in the admin panel. If you are used to Wordpress, look for integrations with the forums listed above, read reviews on how everything works. And be more specific about what you need.

Googled in a minute smashfreakz.com/2012/10/wordpress-forum-plugins/

Of the free ones, I can recommend FUDForum

So I found this. Chose 2: BBPress or PHPBB
I'm wondering who did it.
Do I need to finish the forum engine after integration?
And how much could it cost?

CMS Drupal has a forum built in by default. True, I have never used it, but I am sure that it can be configured, and there are probably a lot of modules for working with the forum. There are a lot of topics for Drupal, you can make different topics for the forum and the main site. CMS is free. The cost depends on the complexity. I think from 5000r, but I could be wrong.

I would still advise looking aside the site-builders.ru constructors , specifically ucoz - the forum and news modules are activated in one click, the best solution in my opinion)

You can create a website using a step-by-step course, he went through it himself. Consists of only 10 video lessons. Everything is explained in detail, so there will be no difficulties. After completing this free course, in 1 hour you will have your own website with all the settings and SEO optimization. Here is the link to the course: lp.partner-80lvl.ru/wpkurs

If you are interested in free website templates, the site https://freehtmlthemes.ru/ provides a wide selection of modern and stylish html templates.