To clarify ... the Tiara and Sweet Sixteen, are more than kissing cousins. HQ makes both machines and then just paints them up for Tiara, and?? .... sorry, there is a 3rd one, but I forget whose it is. I am thinking either Pfaff or Husqvarna. Without a doubt someone will answer this.
Another suggestion ... call the HQ 1-800 Hotline #. Their customer service is exceptional, and they will actually guide you step-by-step in front of your machine. Having a portable handset with a speaker can be very handy.
You mentioned that the original owner gave you extra bobbins. Were these the pre-wound disposable bobbin types? If so, I would recommend going just with the metal ones until you know things are working OK. As already suggested, ensure you have the correct ones for your machine. Also, that they are the Tiara branded bobbins. This way you have eliminated a lot of issue-causing possibilities.
As for thread, you can use any kind of thread in it ..... even as fine as 100 wt right up to quite coarse threads. To answer your question about spool size .... any size is fine. I have used a lot of Aurifil 50 wt., both the small and large for a regular machine, as I wanted to downsize my thread inventory before I built up more. Just not as convenient as the super-sized cones, as they soon empty out when quilting. You travel a lot of miles!!!
You have been given a lot of other really good suggestions in the previous posts, so I will not repeat. I have found when I get into problems, it is often because I just did something wrong in the threading. A key, as others mentioned, make sure your thread is flossed right into the discs .... it is easily thought that it is, when it is not.
Good LUck!