(See the Resources page for more information and downloadable help documents.)
Illumina instruments can read barcodes and indexes. There are 2 common ways this works.

The "Illumina-style" of barcoding/indexing is to insert the index sequence into the adapter and perform an additional sequencing step to prime and read that sequence. This will cost more since there is an additional sequencing read. It will also take slightly longer for the Core to demultiplex the data before delivery.
If you use the adapter-based indexes with the additional sequence read, you must provide the Core with a list of sample names and index sequences in order for us to sort them. If making your own adapter oligos, we recommend using the index sequences provided in support documents from Illumina. The Core STRONGLY recommends using >=8bp Dual or Unique Dual indexes for very-high-throughput instruments such as the NovaSeq to ensure all the indexes can be sorted cleanly.

For an "inline" barcode, you can add an index or barcode to your sequence when you build. This is generally added to the 3' end of the upstream adapter (next to the insert) so that this is the first 4-6 bases read during the sequencing run. Then you sort the data by these first bases into your groups.  If you do this, keep them to 6 or less bases and design them to be as clear as possible.
We recommend that you design in such a way that if one base were misread or the first base was missed, you could still tell which barcode it is. You can also look at commercially available or previously published lists and use those. Additionally, please design a sequence that is NOT part of the sequencing primer or part of your PCR primers, for obvious reasons.

In any style of index, please design and mix to provide the least base overlap (to prevent mis-matches) and the most base diversity.