Python 3.8 (pre-release),  425  423 bytes

-2 bytes by using a splat inside str.replace. Source.

Based on J Atkin's answer—go upvote that one.

a=input()+'=';b=d=q=0;c="0123456789abcdefghijklmnopqrstuvwxyz-+^=";e=[0,1];f,g='+^';h=[i==' 'or'@'<i<'['for i in a];i=abs;p=lambda z:h[q:q+z]==[b]*z
def l(z):global s;k=c.index(z.lower().replace(*' -'));m,n=s=[k%10,k//10];m-=e[0];n-=e[1];return min(i(m),i(10-m))+min(i(n),i(4-n))
def o(z):global d,e;d+=l(z);e=s
for r in a:
 if p(3):o(g);b^=1
 if p(2):
  if l(f)<l(g):o(f)
  else:o(g);b^=1
 if p(1):o(f)
 o(r);q+=1
print(d)

Try it online!

Python 679 661 619 602 589 576 539 520 496 482 Bytes

Run this and it will ask for a input (without prompt text). For the input Code Golf it prints 43.

a=input()+'=';b=0;c="0123456789abcdefghijklmnopqrstuvwxyz-+^=";d=0;e=[0,1];f='+';g='^';h=[i.isupper()or i==' 'for i in a];i=abs;p=lambda z:all([i==b for i in z]);q=0
def l(z):global s;k=c.index(z.lower().replace(' ','-'));s=[k%10,int(k/10)];m,n=s;return sum([min(i(m-e[0]),i(10-(m-e[0]))),min(i(n-e[1]),i(4-(n-e[1])))])
def o(z):global d,e;d+=l(z);e=s
for r in a:
 if p(h[q:q+3]):o(g);b^=1
 if p(h[q:q+2]):
  if l(f)<l(g):o(f)
  else:o(g);b^=1
 if p([h[q]]):o(f)
 o(r);q+=1
print(d)

Full program:

input = input() + '='
capsOn = False

keys = "0123456789abcdefghijklmnopqrstuvwxyz-+^="
totalKeys = 0
caret = [0, 1]

shiftKey = '+'
capsKey = '^'

cases = [char.isupper() or char == ' ' for char in input]

def locate(char):
    """
        Find the location of the char on the keyboard
        regardless of case
    """
    location = keys.find(char.replace(' ', '-').lower())
    return [location % 10, int(location / 10)]


def dist(key):
    """
        Calculate the min dist to a char
    """
    nx, ny = locate(key)
    return sum([min(abs(nx - caret[0]), abs(10 - (nx - caret[0]))), min(abs(ny - caret[1]), abs(4 - (ny - caret[1])))])


def moveTo(char):
    """
        Move the caret to the char, ignoring case and
        adds the dist to the tally
    """
    global totalKeys, caret
    totalKeys = totalKeys + dist(char)

    print(keys[caret[0] + caret[1] * 10], '->', char, '=', dist(char))

    caret = locate(char)

diffCase = lambda case: all([i == capsOn for i in case])

for ind, ch in enumerate(input):
    if diffCase(cases[ind:ind + 3]): # use caps
        moveTo(capsKey)
        capsOn ^= 1
    elif diffCase(cases[ind:ind + 2]): # use closest
        if dist(shiftKey) < dist(capsKey):
            moveTo(shiftKey)
        else:
            moveTo(capsKey)
            capsOn ^= 1
    elif diffCase([cases[ind]]): # use shift
        moveTo(shiftKey)

    moveTo(ch) # apply the move

print('Total:', totalKeys)

Extended output from the full program:

Code Golf
a -> C = 2
c -> ^ = 6
^ -> o = 5
o -> d = 2
d -> e = 1
e -> + = 5
+ -> _ = 1
- -> + = 1
+ -> G = 3
g -> o = 3
o -> l = 3
l -> f = 5
f -> = = 6
Total: 43

Saved a byte thanks to @justin https://codegolf.stackexchange.com/a/18983/42736
4 @xnor https://codegolf.stackexchange.com/a/40791/42736 19 thanks to @Alex

Ruby (369 bytes)

Takes input from command line.

K="0123456789"+('A'..'Z').to_a.join+" +^="
Q=K.downcase.sub' ','-'
def d x,y
t,s=(x/10-y/10).abs,(x%10-y%10).abs
[t,4-t].min+[s,10-s].min
end
def v s,i,l,a
return l if s.empty?
c,r=s[0],s[1..-1]
j=K.index(c.upcase)||36
return v(r,j,l+d(i,j),a)if a.include?c
s,p=d(i,37)+d(37,j),d(i,38)+d(38,j)
[v(r,j,l+s,a),v(r,j,l+p,a==K ? Q : K)].min
end
puts v("#{ARGV[0]}=",10,0,K)

Saved a bunch of bytes thanks to @Charlie :)

Swift 1.2, 812 588 670 bytes

Edit: Removed 224 bytes by replacing the large arrays of numbers with a Range and converting it to an Array instead.

Edit2: Added looping vertically

typealias S=String
typealias I=Int
var A:(I)->S={S(UnicodeScalar($0))},B:(I)->(I,I)={a in(a%10,a/10)},a=Array(48...57).map{A($0)},b=[a+(Array(65...90)+[32,43,94,61]).map{A($0)},a+(Array(97...122)+[45,43,94,61]).map{A($0)}],z=Process.arguments
z.removeAtIndex(0)
func C(e:I,f:I)->I{let(a,b)=B(e),(c,d)=B(f)
return min(abs(d-b), abs(4-(d-b)))+min(abs(c-a),abs(10-(c-a)))}
func D(c:S,_ e:I=10,_ f:Bool=false,_ g:Bool=false)->I{if count(c)==0{return C(e,39)}
let h=c.startIndex,i=c.endIndex,j=S(c[h])
if let k=find(b[f ?1:0],j){return C(e,k)+D(c[advance(h,1)..<i],k,(g ?(!f):f),false)}else{return min(C(e,37)+D(c,37,!f,true),C(e,38)+D(c,38,!f,false))}}
print(D(" ".join(z)))

To run, put the code in a .swift file and run it with swift <filename> <your name>

This uses the simple approach where the two 'keyboards' are stored as arrays.

B:(I)->(I,I)={a in(a%10,a/10)} Converts an index from the array to an x,y position on the virtual keyboard.

func C(e:I,f:I)->I{let(a,b)=B(e),(c,d)=B(f) return abs(d-b)+min(abs(c-a),abs(10-(c-a)))} Takes a start/end index and returns the minimum number of moves to get from once to the other (accounting for horizontal wrap)

func D(c:S,_ e:I=10,_ f:Bool=false,_ g:Bool=false)->I Is the main recursive function that does most of the calculations. It calculates the distance from the current position to the target character, unless the case should change, then it calculates both the shift and the caps lock methods and takes the smallest.

Running swift codegolf.swift Code Golf prints 43

C 675 Bytes

Takes input from command line argument. Uses recursive main:

#define Y(_) (!isdigit(_)?!isalpha(_)?3:1+(toupper(_)-65)/10:0)
#define X(_) (!isdigit(_)?!isalpha(_)?_-32&&_-45?_-43?9-(_==94):7:6:(toupper(_)-5)%10:_-48)
x,y,z;char*s;main(a,_,p,q,r){a<2?s[_]?!isdigit(s[_])&&((s[_]-32&&!isupper(s[_]))||!a)&&((s[_]-45&&!islower(s[_]))||a)?q=x,r=y,main(3,43),p=z,x=X(43),y=Y(43),main(3,s[_]),p+=z,x=X(s[_]),y=Y(s[_]),main(a,_+1),p+=z,x=q,y=r,main(3,94),q=z,x=X(94),y=Y(94),main(3,s[_]),q+=z,x=X(s[_]),y=Y(s[_]),main(!a,_+1),q+=z,z=(q<p?q:p):(main(3,s[_]),q=z,x=X(s[_]),y=Y(s[_]),main(a,_+1),z+=q):(main(3,61)):(a<3?s=((char**)_)[1],x=0,y=1,main(1,0),printf("%d",z):(x=X(_)-x,y=Y(_)-y,x+=10*(x<0),y+=4*(y<0),z=(x>5?10-x:x)+(y>2?4-y:y)));}