So far just looking at reviews and also taking into account games that just appealed to me for various reasons I am thinking about either Alpha Centauri or Heroes of Might and Magic (the first one since if I like it, I'd want to play them all over time) for a TBS and for RTS I am not really sure except I noticed a lot of people really like the classic Total Annihilation but is it overly complex for a new player?
I would not really recommend the first Heroes of might & magic. It is a historical curiosity, but the other games in the series are just so much better that it becomes a bit pointless. Don't worry, you are not missing out much in the way of story. I think that Age of Wonders 1 is better than Heroes of might & magic, so you might be interested in checking out that game as well. Alpha Centauri is a complex game, but if you play on the lowest difficulty it will turn off a lot of the things that makes the game hard to get into for beginners. As you increase the difficulty more and more things will be added that makes the game more complex. I wish other 4X games did this, because it makes the game a bit more gentle for beginners. Expect to spend some time with this game before you really get into it, but it is a great game well worth the time you invest.
As for RTS, Total Annihilation might not be the easiest RTS to get into, but it is a very good game, so as long as you spend a bit of time with it, I'm sure you will enjoy it. It is considered one of the best games of its particular type of RTS. Once you are familiar with RTSs I would recommend checking out Dark Reign (this is not a beginners RTS though).
There are many different types of strategy games, even within the genres of RTS & TBS so if you don't like your first game, don't give up on the genre altogether. Dungeon Keeper for an example is very different from Total Annihilation, even though it is an RTS, as is Ground Control. On the TBS side you have games like Jagged Alliance which offers a more tactical experience, with a focus on the individual soldiers, while a game like Master of Orion will let you be the emperor of a galaxy spanning empire, where you don't have to worry about the minor details but rather the bigger picture (unless you try to play Master of Orion 3, which can result in detail overload).