Plotting a scatter plot in python3 where x axis is latitude/longitude in km and y axis is depth

For the distance calculation you can to use the geopy package, specifically geopy.distance.geodesic(), to calculate the distance along an arc by assuming a particular ellipsoid (e.g. WGS84).

To generate a plot similar to what you've described you can use the matplotlib library's scatterplot functionality, specifically matplotlib.pyplot.scatter().

The code example below will step you through both the distance calculation (distance from some reference lat/long to another lat/long... this isn't necessarily the N-S component but it's easy enough to calculate). As well as how to generate the scatter plot using your Group_ID field to colour the points using two methods.

import matplotlib.pyplot as plt
import geopy
import pandas as pd

# Load your sample data to a Pandas DataFrame where each column corresponds to
# 'Latitude', 'Longitude', 'Depth_m', 'Sample_ID', 'Group_ID'
datafile = r'<path to a file containing your data>'
df = pd.read_csv(datafile)

# Defining one end of our arc to calculate distance along (arbitrarily taking 
# the first point in the example data as the reference point).
ref_point = (df['Latitude'].iloc[0], df['Longitude'].iloc[0])

#  Loop over each sample location calculating the distance along the arc using
#  pygeo.distance.geodesic function.
dist = []
for i in range(len(df)):
    cur_point = (df['Latitude'].iloc[i], df['Longitude'].iloc[i])
    cur_geodesic = geopy.distance.geodesic(ref_point, cur_point)
    cur_dist = cur_geodesic.km
    dist.append(cur_dist)

# Add computed distances to the df DataFrame as column 'Distance_km'
df['Distance_km'] = dist

# Create a matplotlib figure and add two axes for plotting
fig = plt.figure()
ax1 = fig.add_subplot(211)
ax2 = fig.add_subplot(212)

# Example 1: of creating a scatter plot using the calculated distance field and
# colouring the points using a numeric field (i.e. Group_ID in this case is numeric)
pts = ax1.scatter(df['Distance_km'], df['Depth_m'], s=30, c=df['Group_ID'], cmap=plt.cm.jet)
plt.colorbar(pts, ax=ax1)

ax1.set_xlabel('Arc Distance from Reference Point (km)')
ax1.set_ylabel('Depth (m)')
ax1.set_title('Colouring Points by a Numeric Field')
ax1.invert_yaxis()
ax1.grid(True)

# Example of creating basically the same scatter plot as above but handling the
# case of non-numeric values in the field to be used for colour (e.g. imagine 
# wanting to the the Sample_ID field instead)
groups = list(set(df['Group_ID'])) # get a list of the unique Group_ID values
for gid in groups:
    df_tmp = df[df['Group_ID'] == gid]
    ax2.scatter(df_tmp['Distance_km'], df_tmp['Depth_m'], s=30, label=gid)
    
ax2.legend(loc='upper center', title='Legend')
ax2.set_xlabel('Arc Distance from Reference Point (km)')
ax2.set_ylabel('Depth (m)')
ax2.set_title('Colouring Points with Using Categorical Values')
ax2.invert_yaxis()
ax2.grid(True)

fig.tight_layout()
plt.show()

And the output figure...

I am not sure what you are trying to with distance, but conceptually you need to get your x output into your dataframe as a new column as I have done.In terms of having a different color for groups, I would use seaborn for this as they have a hue parameter. Please see the output below of your first scatterplot and an attempt at what you are trying to do with your second scatterplot:

import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
from math import radians
from sklearn.neighbors import DistanceMetric
import seaborn as sns
fig, ax = plt.subplots(nrows=2)
sns.scatterplot(data=df, x='Latitude', y='Depth_m', hue='Group_ID', ax=ax[0])
df['Latitude'] = np.radians(df['Latitude'])
df['Longitude'] = np.radians(df['Longitude'])
dist = DistanceMetric.get_metric('haversine')
df['Distance'] = (dist.pairwise(df[['Latitude','Longitude']].to_numpy())*6373)[0]
sns.scatterplot(data=df, x='Distance' , y='Depth_m', hue='Group_ID', ax=ax[1])
plt.show()