I'm looking for new manga to read. As said in title, I'm looking for manga where they "die" or something happens in there previous life and then they are "reborn" in another world. Also, they keep memories of who they were in the past.

Please no horror because I totally suck at reading those types. A little bit of romance would be nice but if there's none, I don't really mind.

A few examples would be:
Koushaku Reijou
From Maid to Mother
Hajimari no Niina

Obsessed with reincarnation
....
Key to the Stars

These seem to be quite popular lately.

Fantasy/Video game inspired:
Kumo Desu ga, Nani ka? (This light novel is super good and much farther ahead)
The New Gate (Technically not reincarnated, but for all intents and purposes it fits)
Kenja no Mago
Tate no Yuusha no Nariagari
Dungeon Seeker
Jinrou e no Tensei, Maou no Fukukan
Bokura no Kiseki
Tensei Shitara Slime Datta Ken
Tenseishichatta yo (Iya, Gomen)
World Customize Creator
Exterminator (only one chapter so far but looks promising)
Death March kara Hajimaru Isekai Kyousoukyoku
Tsuki ga Michibiku Isekai Douchuu
Re:Monster
ARIFURETA SHOKUGYOU DE SEKAI SAIKYOU (this one is more violent then the others)
Threads of Time

More romance type:
Please Save My Earth
Kanata Kara (Not reborn, but transported)
Sora wa Akai Kawa no Hotori (Same as Kanata kara)

Does the walker choose the path, or the path the walker?

Koushaku Reijou no Tashinami deals with a businesswoman who gets transported to a medieval world and is low-key reincarnated as a pretty important person. She keeps her memories and uses the 21st century knowledge to her own benefit.

Youjo Senki was a pretty popular anime last season. The manga is exactly the plot you are looking for.

Currently getting my life together~ (unsuccessfully)

From maid to mother