Its not wet lab work, but there is ENORMOUS amounts of biological/genomic data at places like the Protein Data Bank [1] and through the NIH[2] (among countless others). Building something like a genome annotator or cross-species DNA/RNA/Protein comparator might not be the most sexy but you'll need some familiarity with these types of tools if you want to do any (successful) gene editing.
Towards wet lab, a lot of the actual reagents (DNA oligos, enzymes, etc.) are not going to be easy to get for personal use (for obvious reasons), but an understanding of how these work is going to be necessary. You'll need to know how to use PCR but you won't be able to get an instrument for under a few hundred dollars on eBay, so you could program/build your own thermocycler (for example using a spare CPU[3]).
If you want to participate in these types of experiments without getting your hands wet, you can also check out the online games Eterna[4] (for RNA design, out of Stanford) or FoldIT[5] (for protein design, out of UWash). I used to be involved in the lab that runs Eterna and some of the stuff they're working on is very cool - mRNA vaccines, artificial ribosomes, and fluorescent molecular sensors.
[1] - https://www.rcsb.org/
[2] - https://www.ncbi.nlm.nih.gov/genome/
[3] - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4165218/
[4] - https://eternagame.org/
[5] - https://fold.it/