Firstly, I gotta tell you that Conan is a heavily episodic series with an amazingly gripping main plot that gets addressed infrequently. However, the fact that this gripping main plot doesn't develop at a fast pace doesn't diminish the experience one bit. The detective work, drama, humor, themes, etc.. of the individual cases is as gripping on a case by case basis. Catching up to Conan was one of the most enjoyable experiences of my life.
The anime in the early days (I'd say the first 200ish) episodes is perfect tonally for the source material. The older grittier style of animation and art adds to the grimy atmosphere of some of the cases in a way that enhances the manga material tremendously. Later on, the animation style gets more generic and colors get brighter that it becomes similar to the manga but in motion form. So given that it's longer than OP lol (1050 chapters), I'd recommend starting with the anime then switching over to the manga.
Here's a great guide for the Conan anime from the conan wiki.
https://www.detectiveconanworld.com/wiki/Anime
The anime has many filler episodes that are marked (TV original) so feel free to skip them they add nothing to the story.
Also 2-3 of the earlier cases are marked (h.a) as in heavily altered. So when you reach those episodes, click on the manga source to get the corresponding manga chapters and read them instead of watching that episode to get the full experience.
I hope I didn't drive you away lol so here's a tl:dr on how to watch
1) Start with the anime.
2) Look at the guide on the wiki skipping the TV original episodes.
3) There're 3 cases that are marked (h.a), read the corresponding manga chapters instead than watching the episodes.
4) Do this for the first 200ish episodes then switch to the manga.
Of course that's all my opinion as a 15 year long fan on how to best enjoy this amazing series. If you love detective stories, you'll get hooked right away. Feel free to ask whatever you want about Conan. It's my life.