Turing machine and coming up with an idea
Just to expand the comments above, you should try to find a high level algorithm, and then refine its steps in smaller pieces.

For example, if you want to recognize palindrome words, a high level agorithm can be:

1. check if the leftmost symbol is equal to the rightmost symbol
2. if they are not equal reject, otherwise delete them
3. if there are other symbols then goto 1, otherwise accept


That can be refined in the following way:

1.1 move the head on the leftmost symbol
1.2 "store" its value
1.3 if it is the only symbol left then accept (w has an odd length)
1.4 otherwise move the head to the rightmost symbol
2.1 compare it to the "stored" one
2.2 if they are not equal halt and reject, otherwise
2.3 delete the rightmost symbol
2.4 move the head to the leftmost symbol
3.1 delete it
3.2 if the tape is empty halt and accept, otherwise
3.3 goto 1.1


The steps are "implemented" using the internal states of the Turing machine. 
While designing the Turing machine you can name the states with labels instead of "q0, q1, q2" to make the design process easier.

If you have a single tape Turing machine, the step [1.2 "store" its value] means that you must use the internal states to "remember" the symbol read while the head moves to the rightmost one.

Suppose you have two input symbols $a,b$ and $-$ is the blank symbol, then you can use a set of states La,L'a,CMPa and Lb,L'b,CMPb to represent (store) the leftmost symbol read; so steps 1.1, 1.2, 1.3, 1.4, 2.1, can be implemented in this way:

     |  a      |   b     |   -
START| a,R,La  | b,R,Lb  | -,R,START &lt;&lt; move right until you find a symbol      
La   | a,R,L'a | b,R,L'a | ACCEPT    &lt;&lt; step 1.3  
Lb   | a,R,L'b | b,R,L'b | ACCEPT    &lt;&lt; step 1.3
L'a  | a,R,L'a | b,R,L'a | -,L,CMPa  &lt;&lt; step 1.4 
L'b  | a,R,L'b | b,R,L'b | -,L,CMPb  &lt;&lt; step 1.4
CMPa | -,L,MVL | REJECT  |           &lt;&lt; compare to a
CMPb | REJECT  | -,L,MVL |           &lt;&lt; compare to b
MVL  | .......                       &lt;&lt; move left


As you can see after reading the leftmost symbol the states are doubled:

Lx we are on the first symbol after the leftmost one (which is symbol x)
L'x we are moving rightward from the leftmost symbol x
CMPx we are comparing the rightmost symbol with the leftmost symbol x