Better for a beginner is claiming central space immediately (e4 /d4, e5/d5) and forget about hypermodern strategies for quite some time.
What is the alternative to keep players like the OP out of trouble? Make them play 1.e4 e5 for both colors and watch them lose a slew of short games until they get it right?
The difference is that you don't see as many tactical themes in closed hypermodern games as opposed to open classical games. I mean, just look at OP's game; he didn't see any tactical themes before placing himself in a near losing position. I love hypermodern openings, don't get me wrong, but they just don't teach tactics as well as a classical opening.
On topic: Re1 is not a usual move for the system you are using. The rook wants to be on f1 to prepare for an f4 break. More usual is to play Qe1 for the e4 break and to prepare for it with c3 (to prevent Nb4). h3 was also not required but that was already explained.
Re: Qe1 vs Re1 -
Since the Queen really belongs on e2 rather than on e1, another possible move order is e3, then Qe2, then e4. This loses a tempo, of course... but so does Qe1, then e4, then Qe2.
Either way, the combination of the moves h3 plus Re1 was unfortunate.
I guess the OP used the generic KIA formula: Nf3, g3, Bg2,0-0, d3. Then you "somehow" manage e4, and get the usual "huge attack".
However, playing e2-e4 under favourable conditions is not trivial at all, and the expected "huge attack" is mostly fictitious, without generous Black cooperation.
The most common scenario in those cases is white being positionally slaughtered right from the opening, and then posting at chess.com and asking silly questions like the OP did.
As is common in such threads, waffllemaster and pfren have offered excellent advice.
I remember getting angry at IM John Donaldson because he wrote in A Strategic Opening Repertoire that players should stick to 1.e4 and 1.d4 until they had reached som level above where I was at the time. I was mid-1400s USCF and in my late 30s. I wanted to play the Reti and the English and was bored with 1.e4 (mostly because I hated playing against the French) and 1.d4 (because I was lost against the Slav).
Alas, I was getting beat at tactics in these "positional" openings. When I returned to 1.e4 and 1.d4, and took up the French as Black, my USCF rating began climbing. It took a few years to reach 1600, but then only three more to reach 1800. Less than three years after 1800, I hit 1982 (shortly after placing 2nd behind Donaldson in our strongest local tournament). I've dropped a bit since then, but I remain among the top players in my city.
So play the French defence to gain elo?
It improved my positional understanding.