How do I make my screenshot ask me what to do next every time?

Try this answer first to get something identical to the old behaviour.

You may create a new keyboard shortcut for the command

gnome-screenshot --interactive

as an alternative.

To do that follow the steps.

1. Open Settings > Devices > Keyboard and scroll down to the bottom.
2. Click on the + symbol. A dialogue should pop up.
3. Enter any name you like in the Name box.
4. Enter gnome-screenshot --interactive in the Command box.
5. Click on the Set Shortcut... button and set any keyboard combination which doesn't cause any conflict, for example Super+Print.
6. Log out and log in again.

Now whenever you press Super+Print a window offering various screenshot options (whole screen/current window/selected area) should appear. Once you take the screenshot it will offer you to choose the location to save the screenshot as in the image in your question.

For convenience you may first change the shortcut for "Save a screenshot to Pictures" in the keyboard shortcut list to something else like Super+Print and assign Print to the custom shortcut you just created.

Also you might want to try using

gnome-screenshot -a -c

This command lets you straight away capture an area and copy the image to the clipboard.

Refer to this answer for an ugly workaround to replicate the old behaviour that lets you choose the folder to save the screenshot after capturing.

Here is one ugly workaround to replicate the old behaviour that lets you choose the folder to save the screenshot after capturing.

Step one

You'll need zenity. If it's not installed, install it first by running

sudo apt install zenity

Then create a script, say text file named prtscr-chooser.sh somewhere, say in your home directory. Add the following lines to the file:

#!/bin/bash
gnome-screenshot -f ~/temporary-screenshot &&
SCREENSHOT=$(zenity --file-selection --save --confirm-overwrite)
mv ~/temporary-screenshot "$SCREENSHOT" ||
rm ~/temporary-screenshot

(Put gnome-screenshot -a in place of gnome-screenshot in the second line to capture an area instead of the whole desktop)

Finally make the script executable (refer to this).

Step two

Assign the script to Print Screen:

1. Open Settings > Devices > Keyboard.
2. Unbind Print Screen, i.e. change or remove the shortcut for 'Save a screenshot to Pictures'.
3. Scroll down to the bottom.
4. Click the '+' symbol. A dialogue should pop up.
5. Enter any name you like in the 'Name' box.
6. Enter full path to the script in the 'Command' box, i.e. in this case enter
   /home/YOUR-USERNAME/prtscr-chooser.sh (your actual username in place of YOUR-USERNAME).
7. Click the 'Set Shortcut...' button and press Print Screen.
8. Log out and log in again.

(You can follow the same method to assign the script the some other keyboard combination, e.g. Shift+Print Screen)

Now you should be getting a 'save as' dialogue which will let you choose the folder and name for the screenshot.

You can get the old behaviour back using the screenshot tool from MATE desktop environment, example:

You should be able to use this on GNOME without an issue.

First install mate-utils by running the following command in Terminal:

sudo apt install mate-utils

Then you would be able to get the old behaviour with the mate-panel-screenshot command. You can use the -a or -w options with this command for capturing an area or the focussed window, respectively, instead of the whole screen.

You can assign the command to PrtScr (or Shift+PrtScr or Alt+PrtScr) following the steps below:

1. Open Settings > Devices > Keyboard.
2. Unbind PrtScr (or Shift+PrtScr or Alt+PrtScr), i.e. change or remove the shortcut for 'Save a screenshot to Pictures' (or 'Save a screenshot of an area to Pictures' or 'Save a screenshot of a window to Pictures').
3. Scroll down to the bottom.
4. Click the '+' symbol. A dialogue should pop up.
5. Enter any name you like in the 'Name' box.
6. Enter the command mate-panel-screenshot in the 'Command' box (or mate-panel-screenshot -a for an area or mate-panel-screenshot -w for a window).
7. Click the 'Set Shortcut...' button and press PrtScr (or Shift+PrtScr or Alt+PrtScr).
8. Log out and log in again.

You can install a Gnome extension called Screenshot tool. Very handy, a dialogue box will pop up asking for copying or saving the screenshot just taken.

The previous behaviour of gnome-screenshot (as illustrated in the question) was press PrtScrn, possibly with modifier keys, to take the shot and then the save-as dialogue would come up.

No solution suggested for gnome-screenshot-3.25.0 replicates exactly that. The only way I managed was to install gnome-screenshot_3.10.1 and use apt hold to stick it there.

Not a "good" solution technically.

sudo apt remove gnome-screenshot

Download the 3.10 package rather than adding the Trusty repos

sudo apt install ./gnome-screenshot_3.10.1-0ubuntu1_i386.deb
sudo apt-mark hold gnome-screenshot

There is no popup [from the hotkey combos] now, but the screenshot tool is still there in Ubuntu 17.10.

You can find it via Activities (in the top left corner) if you type a letter or a few letters, s, sc, scr ...

Click on the camera icon, and you get the screenshot app window.

For Ubuntu 20.04:

1. Open Settings -> Keyboard Shotcuts,
2. Press MouseLB at Shift + Print shotcut,
3. Press Backspace button,
4. Scroll to the end of the list,
5. Press MouseLB at + button,
6. Enter anything in Name field,
7. Enter /usr/bin/gnome-screenshot --interactive to Command field,
8. Press MouseLB to Set Shotcut ... button,
9. Press Shift + PtrSc button,
10. Press Add button.

This script:

1. Take screenshot of an area
2. Copy it to clipboard (automatically)
3. Show image
4. Let you save it wherever you want

Step 1:

Create prtscr-chooser.sh file

gnome-screenshot -a -f ~/temporary-screenshot                    #take screenshot of an area
xclip -selection clipboard -t image/png -i temporary-screenshot  #copy to clipboard 
eog ~/temporary-screenshot                                       #open figure
SCREENSHOT=$(zenity --file-selection --save --confirm-overwrite) #get path to save
mv ~/temporary-screenshot "$SCREENSHOT"".png"                          #save

Let it executable:

chmod +x prtscr-chooser.sh

Step 2

Assign the script to Print Screen:

1. Open Settings > Devices > Keyboard.
2. Unbind Print Screen, i.e. change or remove the shortcut for 'Save a screenshot to Pictures'.
3. Scroll down to the bottom.
4. Click the '+' symbol. A dialogue should pop up.
5. Enter any name you like in the 'Name' box.
6. Enter full path to the script in the 'Command' box, i.e. in this case enter
   /home/YOUR-USERNAME/prtscr-chooser.sh (your actual username in place of YOUR-USERNAME).
7. Click the 'Set Shortcut...' button and press Print Screen.
8. Log out and log in again.

(You can follow the same method to assign the script the some other keyboard combination, e.g. Shift+Print Screen)

Take a look at Shutter, it do very well. Screen, window, or area of screen screenshots with various options. Sit in tray and waiting for click or keyboard shortcut. You can rename file or configure auto naming. Here are screenshot of main options window.