Tandem Duplications may show up clearly in your alignments as either mismatches or large insertions, depending on your aligner's techniques.
Fusions may be a bit tricker to spot by eye, but fusion sites are often soft or hard clipped at the break point by the aligner for all spanning reads. It may look like a steep drop in coverage and the individual reads that you click on would show the clipping.
The evidence for fusion genes is almost always based on the paired-end alignments being one on each gene, and we don't provide a lot of support for visualizing that directly. If you go the gear menu in the pile-up plot and select to stack with paired-end matching, you will see pair-end reads that have their mate in a different chromosome or very far away in the same chromosome drawn with an arrow pointing towards their pair.
Another trick useful for this type of investigation is that you can open a second window on the same project from the Window menu. The second window can navigate to the second gene in the fusion event and by putting the windows side-by-side you can investigate the fusion.
But by the sound of your question, you might be searching for algorithms to detect tandem duplications and fusion events. If that's the case, I'm afraid GenomeBrowse won't be your final solution. GenomeBrowse is purely a visualization tool for existing data. If you have detected events, it's great at investigating their quality and genomic context, but it's not an analysis tool.