What exactly are genes, DNA and chromosomes. How do they relate to each other and what is their function?
Your understanding of DNA is correct. DNA is a heteropolymer of monomers called nucleotides and each nucleotide is made up of a sugar (deoxyribose), phosphate and a "base" (A, T, G, C).

As you already know, DNA is organized as a double helix with the Watson-Crick pairing rules. 

A chromosome contains a single DNA double helix. Many organisms have multiple chromosomes which are very different in their DNA sequence. In diploid cells (such as the somatic cells in humans), each chromosome has two copies. The two copies are almost identical and are called "homologous chromosomes". As you know each parent contributes to one of the homologous chromosomes. 

During eukaryotic cell division, each chromosome is duplicated. The duplicates (or sister chromatids), are joined at a region which is called the centromere. Centromere is an attachment point for spindle fibres that pull the sister chromatids apart.

Each chromosome has multiple genes. A gene is a region of DNA that is transcribed i.e. it can give rise to an RNA (which can either be a mRNA that in turn codes for proteins or a non-coding RNA). There are other regions of DNA that are not transcribed and these are generally referred to as intergenic regions. In many eukaryotic genes some parts of the transcribed RNAs are removed as a result of splicing. The retained parts are called exons and the removed parts are called introns. 

Some organelles like mitochondria and chloroplasts also have a unique genome within themselves. A genome is the entire set of chromosomes including the homologous chromosomes and the organellar chromosomes.

However, when talking about a reference genome, which is just a bioinformatic information and not an actual genome, we just have one sequence for each unique chromosome (for example, in humans, chromosomes 1-22, X-chromosome, Y-chromosome and the mitochondrial DNA). A reference genome is used as a basal set and in a population genomics study, differences from the references are highlighted instead of presenting the entire sequences.


  
  Picture source: https://cnx.org/contents/9TxHOD3O@4/The-Nucleus-and-DNA-Replicatio  Note that paired chromosomes (X shape) are found only during replication. Usually the chromosome is like a single thread. Chromatin is the name of the assembly (for e.g., we don't say chromatin-21). 




Bottomline: We have 46 chromosomes then it means we technically have 92 DNA sequences. However you should consider the double helix as one entity (or a metamolecule) because the strands always stay together. For more details you should read a book. Genes by Benjamin Lewin and Molecular Biology of the Gene by James Watson et al. are good books on this topic.