Biotechnology is rapidly advancing and becoming cheaper and easier to use. Genetic engineering once required a well funded research lab. Now genetic modification can be done as an undergraduate lab or even at home. There are many things that can all be done at home if you have know how and the right equipment, but if you are just starting out you will need:
You can find a used one on ebay for under $100 (be careful if you go on ebay some seller won't sell to the general public and others will sell broken goods) but you can find pretty good ones new for around $300-500 if you are just looking for something basic. You could also make one yourself there are lots of simple incubator designs online.
>A pressure cooker or autoclave
A pressure cooker will work for many applications at home, but if you really need something sterile you will need an autoclave. A good autoclave will cost you a couple hundred dollars.
You can buy specialty made plastic plates online or you can use a container made of polypropylene. (The container will be marked with a 5 in a recycle symbol) Polypropylene will survive an autoclave or a pressure cooker
>agar and/or nutrient broth
Different organisms will require different nutrients. So you will need different growth media for different projects. You can find different recipes here. https://wiki.bugwood.org/Diagnosticians_cookbook Please keep in mind that 99% of bacteria are considered "unculturalable" and many may require special conditions to survive on a culture.
Now depending on what you want to culture you will need a different microscope. If you are fine with just culturing fungi and other eukaryotes then you are fine with just a 400x microscope. If you want to culture prokaryotes (bacteria and archea) then you will need a microscope with at least 1000x magnification. A GOOD new brightfield microscope will run you about $300. I recommend a brightfield microscope for beginners as they are easy to use. You may need an oil immersion lense to see bacteria. If you get one of these becareful with it as the lense alone will cost between $50-100 and it can easily be scratched so you will need special whipes for it. You will need immersion oil to use it, but you might be able to get away with coconut oil. (https://lpulaguna.edu.ph/wp-content/uploads/2018/12/1-COCONUT-OIL-AS-AN-ALTERNATIVE-FOR-IMMERSION-OIL.pdf) No matte what you use make sure to clean it after every use.
In order to see bacteria or improve the viability of some cells you will need a stain. You can use malachite green or methyline blue for simple stains and they can be found either at a pharmacy or an aquarium supply store. Other staining reagents will have to be purchased online. Stains can be very helpful in identifying your organism. You can find many of them on amazon. Here is a small list of stains: https://bio.libretexts.org/Bookshelves/Ancillary_Materials/Laboratory_Experiments/Microbiology_Labs/Book%3A_Laboratory_Exercises_in_Microbiology_(McLaughlin_and_Petersen)/03%3A_Preparation_of_Bacterial_Smears_and_Introduction_to_Staining/3.04%3A_Summary_of_Common_Bacterial_Staining_Techniques