Moore, who is 6ft 5ins, scored 10 goals in 21 league appearances this season for Rovers and Torquay, where he had a brief loan spell earlier this term.
The 24-year-old has penned an 18-month deal at Portman Road.
Spence, 26, has signed to the end of the season having been without a club since leaving MK Dons in the summer.
The former West Ham defender played 80 times for the Dons after joining on a full-time contract, following two separate loan spells.
Both are unavailable for Ipswich's FA Cup third-round replay at Lincoln on Tuesday as they were not registered before the original tie.
Ipswich Town boss Mick McCarthy confirmed on Saturday to BBC Radio Suffolk he was still searching for another striker despite signing Moore, having previously had an approach turned down by Preston for Jordan Hugill.
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Ipswich Town have signed striker Kieffer Moore from National League side Forest Green for an undisclosed fee and former MK Dons defender Jordan Spence.