Solid dish soap is a great way to reduce plastic waste in your home and is also quite economical. I always keep a bar by the sink with a dish brush for hand washing (please don't try to use it in your dishwasher!)
As with most cleaning tasks, hot water works best. I wet the dish brush and then scrub it over the bar a few times to load it up with soap. Then I scrub the dirty item as I normally would, rinse and put on the drying rack.
If you prefer the sink or basin full of soapy water method of dish washing, then you can scrub the bar with your dish brush under the tap as you fill the sink, until you reach the desired level of bubbliness.
What you keep your dish soap bar in can be as simple or fancy as you like. I keep mine in an old ceramic canning jar. When the bar gets low, I pull out the old pieces, put a fresh bar in the bottom and put the pieces on top. For illustrative purposes in this post I've put the bar on one of my homemade willow soap dishes, which usually holds hand soap. By our washing machine, I have a bar ready to use as a stain stick which sits on an old peanut butter jar lid.
Tips for greasy dishes
As you might expect from someone who makes salves and balms as part of their business, I deal with some very greasy dishes on a regular basis. I also have a septic system, so if I clog our pipes with gobs of lard and beeswax, it's very much MY problem.
My three best allies for cleaning grease are hot water, soap and washing soda. Washing soda is sodium carbonate (Na2CO3), not to be confused with baking soda which is sodium bicarbonate (NaHCO3). If you managed to retain enough high school chemistry to wonder why the names of these compounds don't follow the usual pattern of chemical formulas (well done, I had to look it up!), then you can read all about it here. The short answer is that these substances were widely used and thus named before chemists knew their actual chemical formulas.
Washing soda is excellent at cleaning grease because it is very alkaline (a pH of around 11, with 7 being neutral and 14 being the most alkaline). Alkaline substances break up fat molecules and make them water soluble, which means they can be washed down the drain without worrying about them building up in your pipes later. Baking soda is also alkaline but it is much weaker, with a pH of only around 8.
Lye is of course the ultimate alkaline substance but it is very corrosive and dangerous to touch, so washing soda is an effective and much safer alternative. Washing soda can still dry out your hands though, so I usually wear gloves when using it. It's also worth noting that lye can be bad for septic systems. If used regularly you will kill off the bacteria that live there to eat the waste, which will cause trouble down the road. This is why you may hear that you shouldn't use drain cleaners in septic tanks, since drain cleaners are usually made of mostly lye. Washing soda is much gentler on the residents of your septic tank.
Washing soda combined with hot or even boiling water will strip the grease right off of your greasiest dishes. For the pots and tools I use to make soaps and salves, I shake in about 1/2 cup of washing soda (or more or less, depending on how much greasy residue there is), add boiling or very hot water and then give it a quick scrub. Once I rinse it, if there's still a bit of grease film, I'll wash it with my solid dish soap and then it's good to go.
Making Washing Soda
Washing soda can sometimes be hard to find in stores (it's usually in the laundry aisle if they have it). If that's the case it is actually possible to make washing soda out of baking soda, which you can learn how to do here. It's very simple and just involves cooking the baking soda in the oven for a bit.
Sources:
https://en.wikipedia.org/wiki/Sodium_carbonate
https://en.wikipedia.org/wiki/Sodium_bicarbonate
https://sciencenotes.org/why-is-baking-soda-called-sodium-bicarbonate/