You need to distinguish between seeing and observing. These are technical words in relativity. And totally distinct things.
Seeing always happens where you are. When there is a distant event, an observation is when you correct the seeing to compute when and where the distant event happened.
So if Alice and Bob are at rest and 8 light years apart then when Alice sees her clock read 2016 she's sees Bob's clock read 2008 but knows that distant event when Bob's clock read 2008 happened back in 2008. Let's also say they each transmit TV shows so we always have signals to detect.
So basically they think their clocks are synchronized and that they are at rest and that they are 8 light years apart.
Now lets say you take off from Alice to Bob at 0.8c. Let's look at what Alice and Bob see and observe.
Alice sees you with her in 2016 and observes it too. Bob sees that in 2016+8=2022 and observes it as having happened in 2022-8=2016.
Later, Bob sees you arrive. His clock reads 2016+(8/0.8)=2026. He observes that too.
Alice sees you arrive at Bob's when her clock says 2016+(8/0.8)+8=2034 but observes it as having happened in 2034-8=2026.
So they see things when light gets to them. But they correct for distance to compute their observations.
Lets look at you. The doppler formula says you get sees stuff Bob move at triple speed and you were seeing Bob 2008 when you left and see 2026 Bob when you arrive. So in your journey you saw 2026-2008=18 years of Bob's TV shows at triple speed. So you aged 6 years.
The doppler formula also says you get TV shows from Alice at 1/3 speed and you were seeing 2016 Alice when you left and see 2026-8=2018 Alice when you arrive. So in your journey you saw 2018-2016=2 years of shows at 1/3 speed. So you aged 6 years. That's consistent.
So they think it took 10 years. You aged only 6 years. The time dilation is 6/10. But you saw Bob age faster than you saw Alice age. You had eight years of backlog episodes you ran through on your way from Alice to Bob.
So you thought Bob was aging slower than you. You thought he aged 6/10 as fast. So the only reason you think you saw him aging faster was because of the episodes in transit that you hadn't seen yet they were still in flight.
If you want know what you observe you'd start with what you see. You saw Bob 2008 age into Bob 2026 in 6 years of your own time. And you saw him move towards you at 0.8c (you agree in the speed since 0.8c is the speed that makes shows arrive at triple speed according to the doppler effect). Since it took 6 of your years for him to get to you at speed 0.8c you think he was 0.8x6=4.8 light years away that's 8(6/10) light years. That's length contraction.
But that's not what you observe. Relativity is about events. There is the event of you seeing the Bob 2008 show. And there is the event of Bob sending it. When you were with Alice you saw (and observed) the Alice 2016 show. And you saw the Bob 2008 show. But what about the event where Bib sent that episode? Where and when do you think he sent it? Basically you've computed that he was 4.8 lightyears away when you got it. But you want to know where he was when he sent it.
So how do you correct for travel time. Alice took the distance to Bob divided it by c and said the signal left that much longer ago. But the distance wasn't changing so that was easier. You think Bob is moving towards you at speed 0.8c and that he will reach you in 6 of your years so is 4.8 lightyears away but he was farther when sent it. If he sent it T years ago then he traveled 0.8cT and the light traveled 4.8lightyears+0.8cT but it moved at the speed of light so 4.8lightyears+0.8cT=1.0cT. So solve 4.8lightyears+0.8cT=1.0cT for T and get $T= (4.8/0.2c)4.8 lightyears. Thats how you correct. So you saw the Bob 2008 episode at the same event as seeing the Alice 2016 episode. But you observed it to be 4.8 light years away and (4.8/0.2)4.8 years ago.
So in your example the things rushing towards you will appear fast and the things rushing away appear slow. That's seeing. And when you correct for travel time you'll think both are running slow. You'll say that the ones moving towards you appeared faster but were really going slower. But you do have a back log of episodes to watch.
You might ask how can they go slower but appear faster. Well remember when you rushed towards your right friend and saw 8+10 years worth of shows? You had a backlog of 8 shows to burn through and ten more to watch When you burn through the backlog and then start getting them at a faster rate than they are sent eventually there you get the episode that they just made. That's exactly when you pass them. In our example it happened after you saw 8+10 years worth of shows.
So you take off in a direction then you see the people in that direction move faster. The people that are closer have less of a backlog so you burn through that backlog and pass them sooner.
Once you pass someone you see them age slower and you see them age slower than you observe them aging slower, this difference creates a backlog. A backlog you could watch later if you turn around and head back to them.
The people behind you already had a backlog and as you take off you see them age at 1/3 speed but observe then aging at 3/5 speed so the backlog grows.