Firstly, let me just say that I am not an expert, (or for that matter completely fluent!) in speaking Welsh. What I am, however, is a veteran of Welsh language courses. Some are better than others, but if I had to say which I thought best, I go for anything by Gareth King.
Also I'm not saying read this and you'll be able to read and write in Welsh. This is intended as a bit of help, a start. The rest is up to you.
Probably the most important thing to remember in learning anything is to keep up with the practice. This is true with learning a language, an instument or driving a car. The more you do it the faster you'll learn. Practice writing letters, or annoy your friends emailing them in Welsh, just keep it up. Practice and practice and.........
 

For Correspondance, Greetings and Farewells

Corrrespondance

Annwyl                -   Dear                           so  'Annwyl John' is Dear John
S'mae?                -   How's things?
Sut dach chi?     -    How are you?
nghyfaill              -   My Friend                  you can say S'mae, nghyfaill? - How's things, my friend?
Diolch yn fawr    -   Thanks                       Diolch yn fawr am dy lythur - Thanks for your letter.
                                                                    Diolch yn fawr am dy e-bost - Thanks for your e-mail
Eiddoch,              -  Yours,                          Eiddoch yn gywir - Yours truely.
Wir?                    -   Really?                      Nage, wir? - No, really!
Dw i'n gweld       -   I see
Iawn, diolch        - Fine, thankyou

Greetings

Helo                                      -        Hello
Bore da                                 -        Good morning
Noswaith dda                        -        Good evening
Nos da                                   -        Good night                       Nos da iwch - Good night to you.
Craig ydy fy enw fi                -        My name is Craig
***** dw i                             -        I am *****                      Craig dw i  - I am Craig
Dyma ngwraig, Patricia        -        This is my wife, Patricia.
Dyma fy merch, Rhiannon   -        This is my daughter, Rhiannon.
Dyma fy mab, Patrick          -        This is my son, Patrick.
Dyma nhad                           -        This is my father
Dyma fy mam                       -        This is my mother
Dyma ngwr                           -        This is my husband
Dyma mrawd                        -        This is my brother
Dyma 'n chwaer                   -        This is my sister
Beth ydy dy enw di?             -        What is your name?
Ble dach chi'n byw?            -         Where do you live?
 

Farewell

Hwyl               -          Bye (Blessings)
Hwyl fawr       -         Goodbye  (Big Blessings!)
Nos da iwch    -         Good night to you!
Da boch chi    -         Goodbye
 

Welsh Colours, Animals & Trees

Colors

Welsh Lliw               English Colour

gwyn                       White
du                            Black
gwrm                       Dark Blue
coch/rhudd              Red
glas                         Green
melyn                      Yellow
brith                        Grey
brown                      Brown
glas                         Blue
gwyrdd                    Green
porffor/glasgoch     Purple
melyn.goch             Orange
melynwyn               Cream
pinc                         Pink
aur                          Gold
ariannaid                Silver
claer                       Transparent
tywyll                      Dark
goleuni                    Light

Animals

Welsh Anifail          English Animal

ci                              dog
cath                          cat
ceffyl                        horse
llygoden                   mouse
cwningen                  rabbit
draenog                    hedgehog
llew                           lion
blaid                         wolf
carw                         stag
danas                       deer
baedd                       boar
mochyn                    pig
buwch                      cow
tarw                         bull
bytheiad                  hound
aderyn                     bird
eryr                         eagle
hebog                      falcon
hwyad                     duck
bran                        raven
alarch                     swan
pysgodyn                fish
eog                         salmon

Trees

Bedwen               birch
Cerdinen             rowan
Gwernen             alder
Helygen              willow
Onnen                 ash
Draenen wen      hawthorn
Derwen               oak
celynnen             holly
afal                     apple
Gwinwydden      vine
Eiddew               ivy
Cawnen              broom
Draennen ddu    blackthorn
Ysgawen             elder
Ffynidwydden     fir
Eithin                  furze
Grug                   heather
Aethen               poplar
Ywen                  yew
 

Grammar

In Welsh, the person (I, you, they, etc) comes after the verb. This is how the verb `to be`, goes in the present tense:

I am                  dw i
you are              wyt ti                       Singular
he is / she is       mae o / mae hi
we are              dym ni
you are             dych chi                     Plural and formal
they are            maen nhw

My         fy
Your      dy
His         ei
Her        ei       (h before a vowel)
Our        ein     (h before a vowel)
Your      eich
Their      eu       (h before a vowel)

Welsh follows the general rule of Verb - Subject - Object, as opposed to the English Subject - Verb - Object:

Gaeth Craig Gwobr      -     Craig got a prize (literally 'Got Craig a Prize')

This is a general rule, except when the 'focus' is one a particular individual or object:

Pwy derrodd y ffenest? CRAIG dorrodd y ffenest.
Who broke the window? CRAIG broke the window.

And promises never to do it again!  :)

Mutations