When chemical reactions take place it is not always obvious to see or sense. Some chemical reactions are done by completely clear liquids that show no visible changes as they exchange atoms. For example, hydrochloric acid and sodium hydroxide:
HCl + NaOH -> H2O + NaCl
Both of these liquids are colorless and once exchanged make water (still clear) and an ionized form of table salt which dissolves in the water so it cannot be seen. If you want to watch the reaction occur you would need some type of indicator that changes color in the presence of an acid or a base. Phenol-red is one example that can change color as something goes from 6.3 or lower pH (acidic HCl) and turns yellow by a pH of 8.0 (basic NaOH). Once the mixture evens out and most of the acid is neutralized by the base the solution will change color. This process is known as a titration and can be used to measure amounts of different solutions in a mixture.
Some reactions are a fair bit more obvious. One example is the thermite reaction where you mix solid Aluminum metal with a solid ferric oxide (Fe2O3). Because these are both solids, they need a fair bit of energy (heat) to get started, but once they do it produces a solid aluminum oxide (Al2O3) and molten iron metal. Extreme heats are given off as the mixture exchanges energy and is a very bright and very hot reaction.
When creating artificial scents and flavors you begin with very strong smelling mixtures. When making bubblegum or mint scents for instance you can begin with an almost pure acetic acid (vinegar) and mix other strong solutions. You know when the two have mixed because the scents will change.
Sometimes when mixing solutions the inputs may be a liquid, solid, or gas and the end result can be a different form of matter completely. This happens when two liquids mix and through a chemical reaction they create a solid precipitate that is left behind after straining out the liquids.
In conclusion, there are many ways you can see a chemical reaction take place. These methods include (but are not limited to) color change, change in smell, change in thermal energy (getting hotter or colder), a change in type of matter (gas, liquid, solid), or a change in pH and other physical qualities. There are many other examples but these are some of the easiest to test for and see without using lots of expensive equipment or other more complicated measures.