Hello Veronica, I've used Silhouette Studio, and it is a little bit different to Illustrator, although the basic functions are the same. Welding didn't make your text into a vector - Silhouette Studio is already a vector program when it comes to text - welding is the same as Illustrator's Pathfinder Merge (to join shapes).
With your design take the stethoscope and pediatric and make a copy - always a good idea to keep originals in case (like me) it all goes terribly wrong.
If this bit of the design in black isn't already all shapes, then use Text>Create Outlines on the letters and if you've used a stroke for the heartbeat line use Object>Path>Outline Stroke.
Now all the parts in black should be shapes, select them all and use Pathfinder>Merge to make one joined shape. Then Object>Compound Path>Make.
With the single shape selected go to Object>Path>Offset Path and enter the number that gives you the desired width gap or cutout. Change the colour to white if you want to see it.
The word nurse will also need to be converted to outlines and Pathfinder>Merge used to make one shape of the letters, followed by Object>Compound Path>Make. This will give one solid shape of the letters, yes all the same colour, colour will be done later.
Make sure your offset shape of pediatric is at the front of nurse with Object>Arrange>Bring To Front. Select both shapes and use Pathfinder>Minus Front to cut out the shape from the word nurse.
Select the word nurse and go to Object>Ungroup, then select all the shapes that make one letter at a time and colour as desired. Group each letter shapes then group the whole word together if you want to.
Hope that helps - I know it looks complicated but once you get the hang of it you'll be fine!