Source of DNA sequences
GenBank and RefSeq have a huge collection of DNA sequences that can be downloaded in Fasta format. GenBank makes it very easy to search for sequences (for example, using the organism name) but will often have redundant data and varied types (everything from whole chromosome sequences to mRNAs to gene sequences to ESTs). RefSeq has much cleaner, non-redundant data but might take a little more effort for you to find the sequences you want. Depending on whether this has any effect on your project, you might want to consider that. On the RefSeq page, under the FTP menu you can click Genomes to see which genomes have RefSeq data available. The Fasta sequences for a particular genome will be stored in a dedicated directory, possibly subdivided into further directories by chromosome. What you want are the Fasta files (ending in .fa, .fasta, .ffn, or something like that), keeping in mind they may be compressed (.fa.gz, fasta.gz, etc).

Alternatively, if you already have a specific organism in mind, then you could do a Google search to see if the genome for that organism has been sequenced. If so, there is almost always a dedicated website where you can download DNA sequences for that genome.