Hi Jenni
Your problems cutting straight lines aren't at all unusual amongst newbies, so don't worry - just practice!
Use a good quality blade rather than one of those available from the DIY sheds - most of us here use brands such as Olsen, Pegas, Flying Dutchman, Pebeo, Niqua... the list goes on and there may be others. Nevertheless, try a brand that other scrollers use successfully and see how you get on. Different blades suit different scrollers so you might have to do a bit of hunting. At least you'll be getting more practice in while you're trying out different blades

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No matter which blade you use, you'll find it has a slight bias. This is because the manufacturing process leaves a slight burr on the right hand side of the blade. As a result, the blade will have a natural tendency to wander slightly to the right. As time passes you'll compensate for this track naturally by feeding wood into the blade at a slight angle, not directly from the front. However, it takes a bit of getting used to when you first start out. It could be that you're having difficulties cutting a straight line because you're presenting the wood straight instead of just slightly at an angle.
When you're cutting, try not to look at the point where the blade meets the pattern line. Instead, look at a point a little further along the line and cut towards it. It's inevitable that you'll wander from the line when you first start using a scroll saw but don't try to get back onto the line too quickly or you'll end up with lots of noticeable bumps. It's a bit like driving a car; learner drivers have a tendency to over-correct their steering.
All wood burns are caused by the friction of a blade which has been in contact with the wood for too long. If you're using a blade which has been cutting for too long and lost its sharpness, it will not be cutting efficiently enough to let it move through the wood quickly. In this case you need to change your blade. Most scrollers find that blades only last 20 - 30 minutes, some even less than this. It could also be that you're using an inappropriate type of blade for the sort of wood you're cutting. A very fine blade with close set teeth on a 44mm piece of wood is always going to struggle, and when a blade doesn't make progress heat builds up, scorching the wood. Another common cause of wood burns is trying to make turns that are too tight for the blade you're using.
Not all scrollers will agree with what I'm going to say, but my own experience has led me to conclude that the biggest enemy of a sharp blade is heat. When a blade gets hot, it quickly loses its cutting edge. This is one of the reasons why I like to operate my saw at a much slower speed than most other scrollers and feed my wood into the blade very gently. I let the blade do the cutting. Surprisingly, my progress isn't much slower than when I saw at high speed and my blades last a heck of a lot longer.
Something else you can do to protect your blades is cover the top of your wood with lo-tack masking tape such as the blue tape which is available from DIY sheds. This tape has a lubricant on its surface (which is why you can unroll the tape) and the lubricant transfers to the blade as you cut through it. It doesn't leave any trace on your wood.
Finally, routers. They can be used either freehand or mounted to a table. When they are mounted to a table, they are normally inverted underneath it so that the cutting bit protrudes upwards through the table. This gives more control over setting the cutting depth. Routers are classified according to their maximum collet size - 1/4", 8mm, and 1/2". The 1/2" routers tend to be too big to be used freehand so they are normally mounted under a table. As with scroll saw blades, the quality of router bits can vary. Don't be tempted by cheap sets of router bits made in China - they have an alarming propensity to break in use and the cutting edge isn't very good either. Instead, use a reputable brand such as
Wealden,
Titman or even
Trend, although Trend is rather pricey.
I'm sure I've missed some details out but hopefully there's enough info here to address those 3 simple questions

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Gill