With adequate preparation, a small investment, and judicious choice of the objective;
6000 peaks are within reach of many people. Particularly if the first experience at altitude has simplified logistics and several options in the same region.
Here are some suggestions of destinations that can meet these criteria:
# 1 The Volcanoes of Bolivia
Bolivia lends itself perfectly to a first experience at altitude without excessive technical difficulty. It is an exotic location with vast deserts, equatorial jungles, volcanoes covered with snow and ice and the Latin culture. Several volcanoes like Parinacota (6348m) or Sajama (6548m) lend themselves well to these iconic first high-altitude climbs. Which involve the technical skills of moving safely on snow slopes and glaciers.
Here is an example: https://www.novo-monde.com/ascension-volcan-parinacota-bolivie/
Northern hemisphere summer (June to September) weather is drier and more stable, so it is the best time to climb in the Andes. And with 4x4 access, most volcanoes can be climbed in one day. But it is advisable to acclimatize with one or more camps. You can also acclimatize
with treks visiting the villages of the highlands between 3000 and 4000 meters where hot springs, lagoons, breathtaking landscapes abound.
# 2 The Cordillera Blanca
Dominated by the Huascaran (6768m) with its flute streaked faces, the Cordillera Blanca is full of possibilities for more technically difficult climbs at 6000 meters. Nevertheless, there is something for all levels.
From Huaraz, the "Chamonix des Andes" you can take a mini-buse called a “collectivo” to one of the valleys surrounded by Pisco (5752m), Tocllaraju (6050m), Chopicalqui (6354m), Artesonraju (6025m) : https://www.youtube.com/watch?v=zSb8EU_kIRg
Once inside the National Park there are many options from single day alpine climbs to extended treks.
With a small budget and bivvy equipment, you can imagine many combinations.
Useful information: it is imperative to take the Cruz del Sur bus line from Lima to Huaraz on time and in shape...
# 3 The Himalayas of Nepal
A must-visit for aspiring mountaineers, the Himalayas is a dream: the cradle of the highest mountains on Earth, and the land of Sherpas, yaks and spirituality.
Near Everest, the Khumbu Valley offers many trekking peaks between 5500 m and 6500 m requiring less cost and logistics than the coveted "giants".
Dependent on distance and technical difficulty, the climbing one day and several days from the base camp, effecting logistics. Spring and autumn are the seasons for this type of climb, avoiding the traditional monsoon.
The simplest peak is arguably Island Peak (6160m) with its incredible panorama of Lhotse, Ama Dablam, and Makalu.
But East Lobuje (6 119m)has an easier ascent in snow (PD). You will find a wealth of useful information on this site: http://project-himalaya.com/info-trekking-peaks.html
These are only suggestions. Many other massifs could be cited and explored. China, the Caucasus or the African continent are full of objectives, sometimes less high but more wild and less known ...
Now it's up to you to build your dreams and write the rest.