I'd like to offer some slightly different advice, particularly since you're going to be going through a lot of completed series: space them out. What I mean is, limit yourself to one episode per day, or only watch an episode or two on weekends... that sort of thing. The reason for that is because it makes the series more meaningful to you if you spend more overall time with it, and make it a part of your routine. One of the benefits of watching new series is that you're forced to wait week by week, and the series ultimately occupies about three to four months of your life. When you have the entire series before you, it can be tempting to watch everything in one or two days. I think one of the major strengths of an anime series compared to, say, a movie, is that having it spaced out makes it have a greater impact on you overall... especially the series endings.
Granted, that style isn't for everyone. It can also take a fair bit of discipline, especially on series that really hype up each following episode. I think you'll savor each series a bit more if you can, though. I think it's fine to watch multiple series concurrently (just don't go crazy - up to six or seven is probably the max I'd be comfortable with), but again, people have different preferences and viewing styles.
Otherwise, I'll echo the words of others and say, watch what ever appeals to you. With time and as you watch a lot of series you'll become more picky, but unless you're a picky person by nature, you'll probably be fine. I've watched a ton of series over the past 11-12 years, and while I went through two or three years when very few series appealed to me, I feel that I can enjoy nearly all series again. The only thing that I'm still burned out on are the "sex comedies" (I hesitate to even call them "romantic comedies"... see series like
Mouse or
UFO Princess Valkyrie, even just watching the openings on YouTube should give you a sense of what I'm talking about). Those are entertaining for the perverted factor, but generally have a weak premise and forgettable characters. If you've seen one, you've seen almost all of them.
There are definitely series that stand out among the years, and you'll usually get those through recommendations or series rankings. Don't be afraid to check out series that aren't as highly talked about, though. You never know - it might be very meaningful to you, but not to everyone else.
Lastly, the overall style of anime has changed over the years. I wouldn't shy away from series that were made in the 90's, but be aware that the pacing, art styles, animation styles, and overall feel will be different from modern series. One major thing to be aware of is that with series from the 90's and early 2000's, there was a sort of "formula" that many series followed. If it was a 24 episode series, the first eight or so episodes were generally devoted to introducing the characters and the world, the rest up to episode 12 created a miniature storyline that showed how the characters interacted, and then the remaining 12 or so episodes represented what the series was
really about. In other words, there was a very slow buildup, and you had to watch at least the first 3-8 episodes to really give the series a fair chance. Series today are a lot better about keeping you engaged from the first episode.
Don't worry about running out of things to watch, and enjoy the journey
