How do I get started on buying kitchen cabinets with a low budget?

Check out habitat for humanity or one of the salvage sites in your area. You never know what you can find. Also check out you local home improvement stores

Here are a few links for you. I would Google to see what you can find locally to save tons of money on shipping costs. Big box home improvement stores have unadvertised bargains on discontinued or slightly damaged (but repairable) returns. Local cabinet makers may have deals too. Research thoroughly before plunking money down for an on line purchase. Good luck!

https://www.thespruce.com/secrets-to-finding-cheap-kitchen-cabinets-1822039

https://www.thertastore.com/kitchen-cabinets/pre-assembled-kitchen-cabinets.html

https://www.costco.com/Semi-Custom-Kitchen-and-Bath-Cabinets-by-All-Wood-Cabinetry-Ships-in-7-10-days-.product.11280458.html

Habitat for Humanity Restore Store is a great place for used items, often including kitchen cabinets for those with a limited budget.

Alternately, you could start with open shelving and add the doors later for all your overhead cabinets.

If you're DIY inclined, using polyurethane on oak finished plywood can make an attractive countertop -- you can cut a 2'x6' piece out of a 4'x8' piece and end up with an "L" shaped countertop for a corner.

First shop around at places like Homedepot or Lowe's. Find businesses in your area that does remodeling. Go to their showrooms, ask questions, get estimates, find out if they take your old cabinets away, ask how many days it will take to install the cabinets. Get everything in writing. Make sure when they say "3 days their consecutively". They love to start the job on Monday on Tuesday they say"need a part have to order it", meanwhile they start another job somewhere else. The job that takes 2 days becomes two weeks. The'll ask for a deposit, only give one third of it. See how long they been in business and check their standing with the Better Business Bureau. Home Advisor is another place to find a contractor, and ask for references. This may seem like a lot to do, but if your spending $1,000.00 or $5,000.00, it's an investment in your home. Good Luck!

If for the most part the bodies of the cabinets are in good shape, holding together, not cracking, not splitting, not mouldy, etc. you can have a "new" look to your kitchen for less money, by keeping the shells, and just buying new doors, rather than buying the complete cabinets. As the doors are the part of the cabinet that gets the most movement, often it is only the doors (or the door hardware) that no longer functions properly on old cabinets.

Any counter top can be taken off, and a new replacement installed easily as long as the base cabinets are level and firmly attached to the wall.

Another way of saving money on cabinets is to eliminate replacing some of, or all of the former upper cabinets. This look gives a modern, airier, more open look to the kitchen as well as saving money, as you are not needing to purchase as many replacement cabinets, or replacement cabinet doors.

If you are very short of storage space then this of course may not be an option, but if you can live with storing less in your kitchen, then you may want to consider going this route. Create a simple pantry in the basement or underused room in the home for extras, seasonal, or seldom used items.

For small items that are used all the time, some people like the look of a couple of floating shelves on the walls instead of the solid more expensive upper cabinets.

Flipturn7 has really good ideas. I would add that if you must replace them, look at a Habitat for Humanity Reclaim store. I've seen brand new cabinets in there for unbelievable prices. Sometimes rich people just have to have newer, and you can benefit.