Addition:

картофель (uncountable) is mostly used "officially", it is rarely heard in everyday speech. We usually say "картошка" instead of "картофель". And when used this way, "картошка" is also uncountable:

Я купил картошку. - I bought potatoes.
Пять килограммов картошки (Genitive Singular!) - 5 kilos of potatoes.
Пирожки с картошкой - patties with potatoes.

It can become countable (I mean "картошка") only when we specify an exact quantity:
Дай мне пять крупных картошек. (Genitive Plural)
However, it is more natural to use "картофелина" in this sense:
Дай мне пять крупных картофелин - is better.

редис (uncountable) is usually substitued by "редиска" in colloquial speech. And it is used as an uncountable noun as well:
Дайте мне килограмм редиски. (Genitive Singular).
When you specify an exact quantity, it becomes countable:
Дайте мне пять редисок. (Genitive Plural).

There are some other nuances.
томат or помидор (a tomato) is countable: два томата (помидора), пять томатов (помидоров), килограмм томатов (помидоров).
But many people tend to say "помидора" (feminine) colloquially, and they use it as a mass noun:
килограмм помидоры, много помидоры, салат с помидорой etc.

However, according to the "official grammar", the word "помидора" is incorrect.

I think, it would be interesting to compose a list of some most common fruits and vegetables, both countable and uncountable.

Fruit.

Фрукт - фрукты (countable).

Countable nouns:

яблоко - яблоки (apple)
груша - груши (pear)
абрикос - абрикосы (apricot)
персик - персики (peach)
арбуз - арбузы (water-melon)
дыня - дыни (melon)
гранат - гранаты (pomegranate)
финик - финики (date)
грейпфрут - грейпфруты (grapefruit)
лимон - лимоны (lemon)
апельсин - апельсины (orange)
мандарин - мандарины (tangerine)
банан - бананы (banana)
кокос - кокосы (coconut)

Uncountable nouns:

вишня (cherries)
черешня (cherries) - someone else has to explain the difference between вишня and черешня, can anyone help? The dictionary says вишня is Prunus cerasus and черешня is Prunus avium
слива (plum) - can be also used as countable: слива - сливы
алыча (cherry-plum)
айва (quince)
клубника (strawberries)
земляника - we do distinguish земляника from клубника, but the species is the same. Can anyone explain?
малина (raspberries)
ежевика (blackberries)
черника (bilberries)
голубика (great bilberries)
клюква (cranberries)
брусника (cowberries)
крыжовник (gooseberries)
смородина (currants)
виноград (grapes)

Vegetables.

Овощ - овощи (countable).

Countable nouns:

томат = помидор / томаты = помидоры (tomato) (Note: colloquial "помидора" is uncountable)
баклажан - баклажаны (egg-plant)
огурец - огурцы (cucumber)
кабачок - кабачки (vegetable marrow)
тыква - тыквы (pumpkin) (I think it can be used as uncountable)
перец - перцы (red or green peppers) - can be also used as uncountable
репа - репы (turnip) - can be also used as uncountable

Uncountable nouns:

капуста (cabbages)
петрушка (parsley)
укроп (dill)
мята (mint)
щавель (oxalate sorrel)
горох (peas)
фасоль (haricot)
картофель, colloquial картошка (potatoes)
морковь (carrots)
свёкла (beet)
редька (black radishes)
редис, colloquial редиска (garden radishes)
хрен (horse-radish)
лук (onion)
чеснок (garlic)

Corrections and/or additions are welcome!