Here I want to give a some impressions of the places where I have been and that are really worth seeing! I hope this short info is helpful and inspiring ;)
Ceahlău is a small but very impressive national park between Gheorgheni and Piatra Neamt. There are white rock walls and towers rising above the steep forests, but on top you can also find meadows with lots of mountain flowers. A description of trails is on the park's website http://www.ceahlaupark.ro/en/trails/. It makes sense to take some more time and to stay overnight at the Cabana Dochia, from where you have many hiking opportunities.
One of the roads to Ceahlău mountains is winding through the gorge "Cheile Bicazului". You can just have a stop by car to stare up those breathtaking high and narrow walls (or buy a kitschy souvenir at the obligatory stands). Or you can even go hiking around them (there is an info board with map at a parking place).
Around Brasov there are many nice and relatively easy reachable places to go for a hike, one of them is Piatra Mare. For example you can start in the village of Timisu de Jos and take the trail through Canionul Şapte Scări (the canyon of seven stairs - you really need them) to Cabana Piatra Mare where you can also sleep (make a reservation, it's small!). But it is not that far to the top, the way up and down is manageable in one day. On good days you can have an amazing view over almost the whole Carpathian arc, because this small massif is situated right at it's bend!
Situated not far from Brasov is also the majestic, sharp-cut ridge of Piatra Craiului. You can get by train to the village Zărneşti at its foot and hike up through beech forest to the Cabana Curmatura. From there many other trails are starting, there is even one along the whole ridge (for more than one day). But keep in mind that in this steep, rocky upper part you are in almost alpine conditions. So you need good equipment (like always actually) and have to be shure-footed. Some trails also have easy climbing passages with steel rope (via ferrata). Not necessary to mention that the panorama up there is stunning...
Crossing the Carpathians from Brasov in direction to Bucharest by bus or train, you simply can't miss Bucegi mountains. At the sides it is probably the most steep and wild massif that I've seen here, but on the top there's a surprisingly gentle and flat plateau. So something for every taste, because you can also get up easily by cable car, for example from Buşteni or Sinaia. But of course it's worth to take more time and effort, and to hike up ;) There are quite many cabanas, for example Caraiman and Babele (above Buşteni). The last one ("old women") is called after one of the famous, strange looking rock formations that are randomly lying in the landscape nearby.
Last but definitely not least (!!) I can recommend you to go to the Fagaras mountains, because there you can experience a real alpine landscape! The easiest way to access this highest mountain chain of Romania is via the Transfagarasan pass road, that is winding up to Balea lake in many turns (and after a tunnel down on the south side). You can get there by bus from Sibiu (https://www.baleabus.ro/en/), for the upper part (from Balea waterfall) you can use a cable car, too. Around the lake there are not only cabanas and souvenir stands, but also some relatively short trails to wonder at the surrounding. If you're up for some more challenge, you can also start on a tour of several days along the ridge. In certain distances cabanas are welcoming you with food and a warm place, but probably you have to use a refugiul or a tent in between sometimes. You can find some planning info here, too: https://www.welcometoromania.eu/Fagaras/Fagaras_Harta_e.htm
For example, we volunteers went to Cabana Podragu (the closest one from Transfagarasan road), on the next day to the highest peak of Romania (Moldoveanu, 2544 m) and back to the road on the third day. Some sections of the ridge trail (red stripe) have difficult passages secured with steel ropes, so inform yourself before (it's usually mentioned on hiking maps with trail description) and better turn around when you don't feel comfortable with it! That's actually true everywhere, so that you can fully enjoy your hiking experiences in the Carpathians :)
Rapsone
salt valley
lake area Sovata
lake Bözöd